Lecture 5: Quantifier Rules

1. Aim of the lecture

We review the formalizations from Lecture 4 and introduce the natural-deduction rules for ∀ and ∃. We first complete examples with functions and nested quantifiers. We then study how to construct and use universal and existential proofs, including the side conditions that make the four rules valid. Finally, we combine quantifiers with the propositional connectives.

Four recurring forms provide a useful review:

variable (Thing : Type)
variable (Student Reads Understands : Thing → Prop)

#check ∀ x : Thing, Student x → Reads x
#check ∃ x : Thing, Student x ∧ Reads x
#check ∀ x : Thing, Student x → Reads x → Understands x
#check ∃ x : Thing, Student x ∧ Reads x ∧ Understands x

2. Functions inside formulas

Terms built with functions can occur inside predicates, relations, and quantified statements.

variable (Thing : Type)
variable (hypatia : Thing)
variable (Student Reads Curious : Thing → Prop)
variable (motherOf : Thing → Thing)
variable (Knows : Thing → Thing → Prop)

#check Curious (motherOf hypatia)
#check ∀ x : Thing, Student x → Knows x (motherOf x)
#check ∃ x : Thing, Student x ∧ Reads (motherOf x)

The parentheses show that we first build motherOf x and then use that term as the argument of a predicate or relation.

3. Nested quantifiers

Quantifier order determines which choices may depend on others.

variable (Thing : Type)
variable (Person : Thing → Prop)
variable (Knows : Thing → Thing → Prop)

-- Every person knows someone.
#check ∀ x : Thing, Person x →
  ∃ y : Thing, Person y ∧ Knows x y

-- Someone knows every person.
#check ∃ x : Thing, Person x ∧
  ∀ y : Thing, Person y → Knows x y

-- Nobody knows every person.
#check ¬∃ x : Thing, Person x ∧
  ∀ y : Thing, Person y → Knows x y

In the first statement, the witness y may change when x changes. The second statement requires one fixed x who knows every relevant y.

4. Universal introduction, `∀I`

Universal quantifier introduction, ∀I

To prove ∀ y : Thing, A y, introduce an arbitrary x. The goal becomes A x. Prove it without using any special property of x; the conclusion then holds for every object in the domain.

-- Universal introduction, ∀I.
-- The universal goal is opened by introducing a new arbitrary thing `x`.
-- Only the first `intro` applies ∀I; the next two introduce implications.
theorem lecture05_forall_intro
    (Thing : Type)
    (Person Curious : Thing → Prop) :
    ∀ y : Thing, Person y → Curious y → Curious y := byThing:TypePerson:Thing → PropCurious:Thing → Prop⊢ ∀ (y : Thing), Person y → Curious y → Curious y
  intro xThing:TypePerson:Thing → PropCurious:Thing → Propx:Thing⊢ Person x → Curious x → Curious x
  intro hXPersonThing:TypePerson:Thing → PropCurious:Thing → Propx:ThinghXPerson:Person x⊢ Curious x → Curious x
  intro hXCuriousThing:TypePerson:Thing → PropCurious:Thing → Propx:ThinghXPerson:Person xhXCurious:Curious x⊢ Curious x
  exact hXCuriousAll goals completed! 🐙

Only the first intro applies ∀I. The following two introduce the assumptions of the implications.

The arbitrariness condition

The object introduced by ∀I cannot be a particular object selected because of a special property. From hHypatiaCurious : Curious hypatia, for example, we cannot conclude ∀ y : Thing, Curious y. After intro x, the goal would be Curious x, whereas the assumption proves only Curious hypatia.

Formally, the variable used in the derivation must not occur free in any open assumption. In Curious x, x is free. In ∀ y : Thing, Curious y, y is bound. Lean enforces this condition through scope: intro x creates a fresh local variable inside the universal goal.

If a source proof deliberately reuses an existing name, Lean may display the older variable as x✝:

x✝ : Thing
hXCurious : Curious x✝
x : Thing
|- Curious x

The dagger only marks automatic renaming. Curious x✝ and Curious x remain different types.

5. Universal elimination, `∀E`

Universal quantifier elimination, ∀E

Given ∀ x : Thing, A x, choose a term t and apply the universal proof to it. This yields A t.

-- A fact about Hypatia does not establish the same fact for every thing.
-- After `intro x`, `hHypatiaCurious : Curious hypatia` cannot close the goal
-- `Curious x`. If a source proof reuses the name `x`, Lean displays the older
-- local variable as `x✝`; the two local objects remain distinct.

-- Universal elimination, ∀E.
-- Every student reads; Marta is a student; therefore Marta reads.
theorem lecture05_forall_elim
    (Thing : Type)
    (Student Reads : Thing → Prop)
    (marta : Thing)
    (hEveryStudentReads : ∀ x : Thing, Student x → Reads x)
    (hMartaStudent : Student marta) :
    Reads marta := byThing:TypeStudent:Thing → PropReads:Thing → Propmarta:ThinghEveryStudentReads:∀ (x : Thing), Student x → Reads xhMartaStudent:Student marta⊢ Reads marta
  have hIfMartaStudentThenReads := hEveryStudentReads martaThing:TypeStudent:Thing → PropReads:Thing → Propmarta:ThinghEveryStudentReads:∀ (x : Thing), Student x → Reads xhMartaStudent:Student martahIfMartaStudentThenReads:Student marta → Reads marta⊢ Reads marta
  have hMartaReads := hIfMartaStudentThenReads hMartaStudentThing:TypeStudent:Thing → PropReads:Thing → Propmarta:ThinghEveryStudentReads:∀ (x : Thing), Student x → Reads xhMartaStudent:Student martahIfMartaStudentThenReads:Student marta → Reads martahMartaReads:Reads marta⊢ Reads marta
  exact hMartaReadsAll goals completed! 🐙

Applying the universal assumption to marta produces Student marta → Reads marta, which can then be applied to hMartaStudent.

The chosen term need not be a simple name. It can be built by a function. If motherOf : Thing → Thing and hEverythingCurious : ∀ x : Thing, Curious x, then hEverythingCurious (motherOf hypatia) proves Curious (motherOf hypatia). Lean checks the term's type and handles bound variables without accidental capture.

6. Existential introduction, `∃I`

Existential quantifier introduction, ∃I

To prove ∃ x : Thing, A x, choose a witness t and prove A t. Lean's constructor is Exists.intro.

-- Existential introduction, ∃I.
-- Choose Marta as the witness and use the assumption that she reads.
theorem lecture05_exists_intro
    (Thing : Type)
    (Reads : Thing → Prop)
    (marta : Thing)
    (hMartaReads : Reads marta) :
    ∃ x : Thing, Reads x := byThing:TypeReads:Thing → Propmarta:ThinghMartaReads:Reads marta⊢ ∃ x, Reads x
  apply Exists.intro martaThing:TypeReads:Thing → Propmarta:ThinghMartaReads:Reads marta⊢ Reads marta
  exact hMartaReadsAll goals completed! 🐙

Choosing marta turns the existential goal into Reads marta, which is already available as an assumption.

The witness can also be a compound term. To prove ∃ x : Thing, Curious x, we may choose motherOf hypatia when we have a proof of Curious (motherOf hypatia).

7. Existential elimination, `∃E`

Existential quantifier elimination, ∃E

Given ∃ x : Thing, A x, we know that a witness exists but may not assume its identity. To derive B, introduce an arbitrary witness y together with A y, and derive a conclusion that does not depend on the identity of y.

-- Existential elimination, ∃E.
-- Someone filed a report; whenever someone files a report, an investigation
-- starts; therefore an investigation starts.
theorem lecture05_exists_elim
    (Thing : Type)
    (FiledReport : Thing → Prop)
    (InvestigationStarts : Prop)
    (hSomeoneFiledReport : ∃ x : Thing, FiledReport x)
    (hReportStartsInvestigation :
      (x : Thing) → FiledReport x → InvestigationStarts) :
    InvestigationStarts := byThing:TypeFiledReport:Thing → PropInvestigationStarts:ProphSomeoneFiledReport:∃ x, FiledReport xhReportStartsInvestigation:∀ (x : Thing), FiledReport x → InvestigationStarts⊢ InvestigationStarts
  apply Exists.elim hSomeoneFiledReportThing:TypeFiledReport:Thing → PropInvestigationStarts:ProphSomeoneFiledReport:∃ x, FiledReport xhReportStartsInvestigation:∀ (x : Thing), FiledReport x → InvestigationStarts⊢ ∀ (a : Thing), FiledReport a → InvestigationStarts
  intro yThing:TypeFiledReport:Thing → PropInvestigationStarts:ProphSomeoneFiledReport:∃ x, FiledReport xhReportStartsInvestigation:∀ (x : Thing), FiledReport x → InvestigationStartsy:Thing⊢ FiledReport y → InvestigationStarts
  intro hYFiledReportThing:TypeFiledReport:Thing → PropInvestigationStarts:ProphSomeoneFiledReport:∃ x, FiledReport xhReportStartsInvestigation:∀ (x : Thing), FiledReport x → InvestigationStartsy:ThinghYFiledReport:FiledReport y⊢ InvestigationStarts
  have hIfYFiledThenInvestigationStarts := hReportStartsInvestigation yThing:TypeFiledReport:Thing → PropInvestigationStarts:ProphSomeoneFiledReport:∃ x, FiledReport xhReportStartsInvestigation:∀ (x : Thing), FiledReport x → InvestigationStartsy:ThinghYFiledReport:FiledReport yhIfYFiledThenInvestigationStarts:FiledReport y → InvestigationStarts⊢ InvestigationStarts
  have hInvestigationStarts :=
    hIfYFiledThenInvestigationStarts hYFiledReportThing:TypeFiledReport:Thing → PropInvestigationStarts:ProphSomeoneFiledReport:∃ x, FiledReport xhReportStartsInvestigation:∀ (x : Thing), FiledReport x → InvestigationStartsy:ThinghYFiledReport:FiledReport yhIfYFiledThenInvestigationStarts:FiledReport y → InvestigationStartshInvestigationStarts:InvestigationStarts⊢ InvestigationStarts
  exact hInvestigationStartsAll goals completed! 🐙

The x in the second assumption is a bound placeholder. We instantiate it with the local witness y, obtaining FiledReport y → InvestigationStarts, and then apply that implication to hYFiledReport.

The conclusion is fixed before the existential is opened and contains no free occurrence of y. Lean enforces this independence through the scope of y and the type of Exists.elim: the witness cannot escape the local derivation unless it is packaged inside another existential statement.

8. Combining the rules

Quantifier rules can be combined in one proof.

-- A more complex example combining ∃E, ∀E, and ∃I.
-- Someone has an umbrella; everyone with an umbrella stays dry;
-- therefore someone stays dry.
theorem lecture05_exists_elim_combined
    (Thing : Type)
    (HasUmbrella StaysDry : Thing → Prop)
    (hSomeoneHasUmbrella : ∃ x : Thing, HasUmbrella x)
    (hUmbrellaStaysDry : ∀ x : Thing, HasUmbrella x → StaysDry x) :
    ∃ x : Thing, StaysDry x := byThing:TypeHasUmbrella:Thing → PropStaysDry:Thing → ProphSomeoneHasUmbrella:∃ x, HasUmbrella xhUmbrellaStaysDry:∀ (x : Thing), HasUmbrella x → StaysDry x⊢ ∃ x, StaysDry x
  apply Exists.elim hSomeoneHasUmbrellaThing:TypeHasUmbrella:Thing → PropStaysDry:Thing → ProphSomeoneHasUmbrella:∃ x, HasUmbrella xhUmbrellaStaysDry:∀ (x : Thing), HasUmbrella x → StaysDry x⊢ ∀ (a : Thing), HasUmbrella a → ∃ x, StaysDry x
  intro yThing:TypeHasUmbrella:Thing → PropStaysDry:Thing → ProphSomeoneHasUmbrella:∃ x, HasUmbrella xhUmbrellaStaysDry:∀ (x : Thing), HasUmbrella x → StaysDry xy:Thing⊢ HasUmbrella y → ∃ x, StaysDry x
  intro hYHasUmbrellaThing:TypeHasUmbrella:Thing → PropStaysDry:Thing → ProphSomeoneHasUmbrella:∃ x, HasUmbrella xhUmbrellaStaysDry:∀ (x : Thing), HasUmbrella x → StaysDry xy:ThinghYHasUmbrella:HasUmbrella y⊢ ∃ x, StaysDry x
  apply Exists.intro yThing:TypeHasUmbrella:Thing → PropStaysDry:Thing → ProphSomeoneHasUmbrella:∃ x, HasUmbrella xhUmbrellaStaysDry:∀ (x : Thing), HasUmbrella x → StaysDry xy:ThinghYHasUmbrella:HasUmbrella y⊢ StaysDry y
  have hIfYHasUmbrellaThenStaysDry := hUmbrellaStaysDry yThing:TypeHasUmbrella:Thing → PropStaysDry:Thing → ProphSomeoneHasUmbrella:∃ x, HasUmbrella xhUmbrellaStaysDry:∀ (x : Thing), HasUmbrella x → StaysDry xy:ThinghYHasUmbrella:HasUmbrella yhIfYHasUmbrellaThenStaysDry:HasUmbrella y → StaysDry y⊢ StaysDry y
  have hYStaysDry := hIfYHasUmbrellaThenStaysDry hYHasUmbrellaThing:TypeHasUmbrella:Thing → PropStaysDry:Thing → ProphSomeoneHasUmbrella:∃ x, HasUmbrella xhUmbrellaStaysDry:∀ (x : Thing), HasUmbrella x → StaysDry xy:ThinghYHasUmbrella:HasUmbrella yhIfYHasUmbrellaThenStaysDry:HasUmbrella y → StaysDry yhYStaysDry:StaysDry y⊢ StaysDry y
  exact hYStaysDryAll goals completed! 🐙

Here Exists.elim opens the first existential, applying hUmbrellaStaysDry to y uses ∀E, and Exists.intro y uses the same object as the witness of the new existential.

Universal quantification and conjunction

Every student who reads takes notes and understands. We introduce an arbitrary student, apply both universal assumptions, and construct the conjunction.

-- Combine ∀I, →I, ∀E, and ∧I.
theorem lecture05_forall_conjunction_combined
    (Thing : Type)
    (Student Reads TakesNotes Understands : Thing → Prop)
    (hReadersTakeNotes :
      ∀ x : Thing, Student x → Reads x → TakesNotes x)
    (hReadersUnderstand :
      ∀ x : Thing, Student x → Reads x → Understands x) :
    ∀ x : Thing, Student x → Reads x → TakesNotes x ∧ Understands x := byThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands x⊢ ∀ (x : Thing), Student x → Reads x → TakesNotes x ∧ Understands x
  intro xThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands xx:Thing⊢ Student x → Reads x → TakesNotes x ∧ Understands x
  intro hXStudentThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands xx:ThinghXStudent:Student x⊢ Reads x → TakesNotes x ∧ Understands x
  intro hXReadsThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands xx:ThinghXStudent:Student xhXReads:Reads x⊢ TakesNotes x ∧ Understands x
  apply And.introleftThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands xx:ThinghXStudent:Student xhXReads:Reads x⊢ TakesNotes xrightThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands xx:ThinghXStudent:Student xhXReads:Reads x⊢ Understands x
  ·leftThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands xx:ThinghXStudent:Student xhXReads:Reads x⊢ TakesNotes x have hIfXStudentThenReadingImpliesNotes := hReadersTakeNotes xleftThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands xx:ThinghXStudent:Student xhXReads:Reads xhIfXStudentThenReadingImpliesNotes:Student x → Reads x → TakesNotes x⊢ TakesNotes x
    have hIfXReadsThenTakesNotes :=
      hIfXStudentThenReadingImpliesNotes hXStudentleftThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands xx:ThinghXStudent:Student xhXReads:Reads xhIfXStudentThenReadingImpliesNotes:Student x → Reads x → TakesNotes xhIfXReadsThenTakesNotes:Reads x → TakesNotes x⊢ TakesNotes x
    have hXTakesNotes := hIfXReadsThenTakesNotes hXReadsleftThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands xx:ThinghXStudent:Student xhXReads:Reads xhIfXStudentThenReadingImpliesNotes:Student x → Reads x → TakesNotes xhIfXReadsThenTakesNotes:Reads x → TakesNotes xhXTakesNotes:TakesNotes x⊢ TakesNotes x
    exact hXTakesNotesAll goals completed! 🐙
  ·rightThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands xx:ThinghXStudent:Student xhXReads:Reads x⊢ Understands x have hIfXStudentThenReadingImpliesUnderstanding := hReadersUnderstand xrightThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands xx:ThinghXStudent:Student xhXReads:Reads xhIfXStudentThenReadingImpliesUnderstanding:Student x → Reads x → Understands x⊢ Understands x
    have hIfXReadsThenUnderstands :=
      hIfXStudentThenReadingImpliesUnderstanding hXStudentrightThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands xx:ThinghXStudent:Student xhXReads:Reads xhIfXStudentThenReadingImpliesUnderstanding:Student x → Reads x → Understands xhIfXReadsThenUnderstands:Reads x → Understands x⊢ Understands x
    have hXUnderstands := hIfXReadsThenUnderstands hXReadsrightThing:TypeStudent:Thing → PropReads:Thing → PropTakesNotes:Thing → PropUnderstands:Thing → ProphReadersTakeNotes:∀ (x : Thing), Student x → Reads x → TakesNotes xhReadersUnderstand:∀ (x : Thing), Student x → Reads x → Understands xx:ThinghXStudent:Student xhXReads:Reads xhIfXStudentThenReadingImpliesUnderstanding:Student x → Reads x → Understands xhIfXReadsThenUnderstands:Reads x → Understands xhXUnderstands:Understands x⊢ Understands x
    exact hXUnderstandsAll goals completed! 🐙

Existential quantification and disjunction

Someone visits Rome or Athens. We open the existential and consider both cases of the disjunction. Each case supplies a witness who sees a museum.

-- Combine ∃E, ∨E, ∀E, and ∃I.
theorem lecture05_exists_or_combined
    (Thing : Type)
    (VisitsRome VisitsAthens SeesMuseum : Thing → Prop)
    (hSomeoneVisitsACity : ∃ x : Thing, VisitsRome x ∨ VisitsAthens x)
    (hRomeVisitorsSeeMuseum : ∀ x : Thing, VisitsRome x → SeesMuseum x)
    (hAthensVisitorsSeeMuseum : ∀ x : Thing, VisitsAthens x → SeesMuseum x) :
    ∃ x : Thing, SeesMuseum x := byThing:TypeVisitsRome:Thing → PropVisitsAthens:Thing → PropSeesMuseum:Thing → ProphSomeoneVisitsACity:∃ x, VisitsRome x ∨ VisitsAthens xhRomeVisitorsSeeMuseum:∀ (x : Thing), VisitsRome x → SeesMuseum xhAthensVisitorsSeeMuseum:∀ (x : Thing), VisitsAthens x → SeesMuseum x⊢ ∃ x, SeesMuseum x
  apply Exists.elim hSomeoneVisitsACityThing:TypeVisitsRome:Thing → PropVisitsAthens:Thing → PropSeesMuseum:Thing → ProphSomeoneVisitsACity:∃ x, VisitsRome x ∨ VisitsAthens xhRomeVisitorsSeeMuseum:∀ (x : Thing), VisitsRome x → SeesMuseum xhAthensVisitorsSeeMuseum:∀ (x : Thing), VisitsAthens x → SeesMuseum x⊢ ∀ (a : Thing), VisitsRome a ∨ VisitsAthens a → ∃ x, SeesMuseum x
  intro yThing:TypeVisitsRome:Thing → PropVisitsAthens:Thing → PropSeesMuseum:Thing → ProphSomeoneVisitsACity:∃ x, VisitsRome x ∨ VisitsAthens xhRomeVisitorsSeeMuseum:∀ (x : Thing), VisitsRome x → SeesMuseum xhAthensVisitorsSeeMuseum:∀ (x : Thing), VisitsAthens x → SeesMuseum xy:Thing⊢ VisitsRome y ∨ VisitsAthens y → ∃ x, SeesMuseum x
  intro hYVisitsRomeOrAthensThing:TypeVisitsRome:Thing → PropVisitsAthens:Thing → PropSeesMuseum:Thing → ProphSomeoneVisitsACity:∃ x, VisitsRome x ∨ VisitsAthens xhRomeVisitorsSeeMuseum:∀ (x : Thing), VisitsRome x → SeesMuseum xhAthensVisitorsSeeMuseum:∀ (x : Thing), VisitsAthens x → SeesMuseum xy:ThinghYVisitsRomeOrAthens:VisitsRome y ∨ VisitsAthens y⊢ ∃ x, SeesMuseum x
  cases hYVisitsRomeOrAthens with
  | inl hYVisitsRome =>inlThing:TypeVisitsRome:Thing → PropVisitsAthens:Thing → PropSeesMuseum:Thing → ProphSomeoneVisitsACity:∃ x, VisitsRome x ∨ VisitsAthens xhRomeVisitorsSeeMuseum:∀ (x : Thing), VisitsRome x → SeesMuseum xhAthensVisitorsSeeMuseum:∀ (x : Thing), VisitsAthens x → SeesMuseum xy:ThinghYVisitsRome:VisitsRome y⊢ ∃ x, SeesMuseum x
      apply Exists.intro yinlThing:TypeVisitsRome:Thing → PropVisitsAthens:Thing → PropSeesMuseum:Thing → ProphSomeoneVisitsACity:∃ x, VisitsRome x ∨ VisitsAthens xhRomeVisitorsSeeMuseum:∀ (x : Thing), VisitsRome x → SeesMuseum xhAthensVisitorsSeeMuseum:∀ (x : Thing), VisitsAthens x → SeesMuseum xy:ThinghYVisitsRome:VisitsRome y⊢ SeesMuseum y
      have hIfYVisitsRomeThenSeesMuseum := hRomeVisitorsSeeMuseum yinlThing:TypeVisitsRome:Thing → PropVisitsAthens:Thing → PropSeesMuseum:Thing → ProphSomeoneVisitsACity:∃ x, VisitsRome x ∨ VisitsAthens xhRomeVisitorsSeeMuseum:∀ (x : Thing), VisitsRome x → SeesMuseum xhAthensVisitorsSeeMuseum:∀ (x : Thing), VisitsAthens x → SeesMuseum xy:ThinghYVisitsRome:VisitsRome yhIfYVisitsRomeThenSeesMuseum:VisitsRome y → SeesMuseum y⊢ SeesMuseum y
      have hYSeesMuseum := hIfYVisitsRomeThenSeesMuseum hYVisitsRomeinlThing:TypeVisitsRome:Thing → PropVisitsAthens:Thing → PropSeesMuseum:Thing → ProphSomeoneVisitsACity:∃ x, VisitsRome x ∨ VisitsAthens xhRomeVisitorsSeeMuseum:∀ (x : Thing), VisitsRome x → SeesMuseum xhAthensVisitorsSeeMuseum:∀ (x : Thing), VisitsAthens x → SeesMuseum xy:ThinghYVisitsRome:VisitsRome yhIfYVisitsRomeThenSeesMuseum:VisitsRome y → SeesMuseum yhYSeesMuseum:SeesMuseum y⊢ SeesMuseum y
      exact hYSeesMuseumAll goals completed! 🐙
  | inr hYVisitsAthens =>inrThing:TypeVisitsRome:Thing → PropVisitsAthens:Thing → PropSeesMuseum:Thing → ProphSomeoneVisitsACity:∃ x, VisitsRome x ∨ VisitsAthens xhRomeVisitorsSeeMuseum:∀ (x : Thing), VisitsRome x → SeesMuseum xhAthensVisitorsSeeMuseum:∀ (x : Thing), VisitsAthens x → SeesMuseum xy:ThinghYVisitsAthens:VisitsAthens y⊢ ∃ x, SeesMuseum x
      apply Exists.intro yinrThing:TypeVisitsRome:Thing → PropVisitsAthens:Thing → PropSeesMuseum:Thing → ProphSomeoneVisitsACity:∃ x, VisitsRome x ∨ VisitsAthens xhRomeVisitorsSeeMuseum:∀ (x : Thing), VisitsRome x → SeesMuseum xhAthensVisitorsSeeMuseum:∀ (x : Thing), VisitsAthens x → SeesMuseum xy:ThinghYVisitsAthens:VisitsAthens y⊢ SeesMuseum y
      have hIfYVisitsAthensThenSeesMuseum := hAthensVisitorsSeeMuseum yinrThing:TypeVisitsRome:Thing → PropVisitsAthens:Thing → PropSeesMuseum:Thing → ProphSomeoneVisitsACity:∃ x, VisitsRome x ∨ VisitsAthens xhRomeVisitorsSeeMuseum:∀ (x : Thing), VisitsRome x → SeesMuseum xhAthensVisitorsSeeMuseum:∀ (x : Thing), VisitsAthens x → SeesMuseum xy:ThinghYVisitsAthens:VisitsAthens yhIfYVisitsAthensThenSeesMuseum:VisitsAthens y → SeesMuseum y⊢ SeesMuseum y
      have hYSeesMuseum := hIfYVisitsAthensThenSeesMuseum hYVisitsAthensinrThing:TypeVisitsRome:Thing → PropVisitsAthens:Thing → PropSeesMuseum:Thing → ProphSomeoneVisitsACity:∃ x, VisitsRome x ∨ VisitsAthens xhRomeVisitorsSeeMuseum:∀ (x : Thing), VisitsRome x → SeesMuseum xhAthensVisitorsSeeMuseum:∀ (x : Thing), VisitsAthens x → SeesMuseum xy:ThinghYVisitsAthens:VisitsAthens yhIfYVisitsAthensThenSeesMuseum:VisitsAthens y → SeesMuseum yhYSeesMuseum:SeesMuseum y⊢ SeesMuseum y
      exact hYSeesMuseumAll goals completed! 🐙

Nested quantifiers

The existential witness may depend on the person introduced by the universal quantifier. We open the existential only after fixing that person.

-- Combine nested quantifiers, ∃E, ∃I, and ∧I.
-- Every person knows at least one person. Therefore, for every person, there
-- is someone whom they know and who is a person.
theorem lecture05_nested_quantifiers_combined
    (Thing : Type)
    (Person : Thing → Prop)
    (Knows : Thing → Thing → Prop)
    (hEveryPersonKnowsSomeone :
      ∀ x : Thing, Person x →
        ∃ y : Thing, Person y ∧ Knows x y) :
    ∀ x : Thing, Person x →
      ∃ y : Thing, Knows x y ∧ Person y := byThing:TypePerson:Thing → PropKnows:Thing → Thing → ProphEveryPersonKnowsSomeone:∀ (x : Thing), Person x → ∃ y, Person y ∧ Knows x y⊢ ∀ (x : Thing), Person x → ∃ y, Knows x y ∧ Person y
  intro xThing:TypePerson:Thing → PropKnows:Thing → Thing → ProphEveryPersonKnowsSomeone:∀ (x : Thing), Person x → ∃ y, Person y ∧ Knows x yx:Thing⊢ Person x → ∃ y, Knows x y ∧ Person y
  intro hXPersonThing:TypePerson:Thing → PropKnows:Thing → Thing → ProphEveryPersonKnowsSomeone:∀ (x : Thing), Person x → ∃ y, Person y ∧ Knows x yx:ThinghXPerson:Person x⊢ ∃ y, Knows x y ∧ Person y
  have hIfXPersonThenKnowsSomeone := hEveryPersonKnowsSomeone xThing:TypePerson:Thing → PropKnows:Thing → Thing → ProphEveryPersonKnowsSomeone:∀ (x : Thing), Person x → ∃ y, Person y ∧ Knows x yx:ThinghXPerson:Person xhIfXPersonThenKnowsSomeone:Person x → ∃ y, Person y ∧ Knows x y⊢ ∃ y, Knows x y ∧ Person y
  have hXKnowsSomeone := hIfXPersonThenKnowsSomeone hXPersonThing:TypePerson:Thing → PropKnows:Thing → Thing → ProphEveryPersonKnowsSomeone:∀ (x : Thing), Person x → ∃ y, Person y ∧ Knows x yx:ThinghXPerson:Person xhIfXPersonThenKnowsSomeone:Person x → ∃ y, Person y ∧ Knows x yhXKnowsSomeone:∃ y, Person y ∧ Knows x y⊢ ∃ y, Knows x y ∧ Person y
  apply Exists.elim hXKnowsSomeoneThing:TypePerson:Thing → PropKnows:Thing → Thing → ProphEveryPersonKnowsSomeone:∀ (x : Thing), Person x → ∃ y, Person y ∧ Knows x yx:ThinghXPerson:Person xhIfXPersonThenKnowsSomeone:Person x → ∃ y, Person y ∧ Knows x yhXKnowsSomeone:∃ y, Person y ∧ Knows x y⊢ ∀ (a : Thing), Person a ∧ Knows x a → ∃ y, Knows x y ∧ Person y
  intro yThing:TypePerson:Thing → PropKnows:Thing → Thing → ProphEveryPersonKnowsSomeone:∀ (x : Thing), Person x → ∃ y, Person y ∧ Knows x yx:ThinghXPerson:Person xhIfXPersonThenKnowsSomeone:Person x → ∃ y, Person y ∧ Knows x yhXKnowsSomeone:∃ y, Person y ∧ Knows x yy:Thing⊢ Person y ∧ Knows x y → ∃ y, Knows x y ∧ Person y
  intro hYPersonAndXKnowsYThing:TypePerson:Thing → PropKnows:Thing → Thing → ProphEveryPersonKnowsSomeone:∀ (x : Thing), Person x → ∃ y, Person y ∧ Knows x yx:ThinghXPerson:Person xhIfXPersonThenKnowsSomeone:Person x → ∃ y, Person y ∧ Knows x yhXKnowsSomeone:∃ y, Person y ∧ Knows x yy:ThinghYPersonAndXKnowsY:Person y ∧ Knows x y⊢ ∃ y, Knows x y ∧ Person y
  apply Exists.intro yThing:TypePerson:Thing → PropKnows:Thing → Thing → ProphEveryPersonKnowsSomeone:∀ (x : Thing), Person x → ∃ y, Person y ∧ Knows x yx:ThinghXPerson:Person xhIfXPersonThenKnowsSomeone:Person x → ∃ y, Person y ∧ Knows x yhXKnowsSomeone:∃ y, Person y ∧ Knows x yy:ThinghYPersonAndXKnowsY:Person y ∧ Knows x y⊢ Knows x y ∧ Person y
  apply And.introleftThing:TypePerson:Thing → PropKnows:Thing → Thing → ProphEveryPersonKnowsSomeone:∀ (x : Thing), Person x → ∃ y, Person y ∧ Knows x yx:ThinghXPerson:Person xhIfXPersonThenKnowsSomeone:Person x → ∃ y, Person y ∧ Knows x yhXKnowsSomeone:∃ y, Person y ∧ Knows x yy:ThinghYPersonAndXKnowsY:Person y ∧ Knows x y⊢ Knows x yrightThing:TypePerson:Thing → PropKnows:Thing → Thing → ProphEveryPersonKnowsSomeone:∀ (x : Thing), Person x → ∃ y, Person y ∧ Knows x yx:ThinghXPerson:Person xhIfXPersonThenKnowsSomeone:Person x → ∃ y, Person y ∧ Knows x yhXKnowsSomeone:∃ y, Person y ∧ Knows x yy:ThinghYPersonAndXKnowsY:Person y ∧ Knows x y⊢ Person y
  ·leftThing:TypePerson:Thing → PropKnows:Thing → Thing → ProphEveryPersonKnowsSomeone:∀ (x : Thing), Person x → ∃ y, Person y ∧ Knows x yx:ThinghXPerson:Person xhIfXPersonThenKnowsSomeone:Person x → ∃ y, Person y ∧ Knows x yhXKnowsSomeone:∃ y, Person y ∧ Knows x yy:ThinghYPersonAndXKnowsY:Person y ∧ Knows x y⊢ Knows x y exact hYPersonAndXKnowsY.rightAll goals completed! 🐙
  ·rightThing:TypePerson:Thing → PropKnows:Thing → Thing → ProphEveryPersonKnowsSomeone:∀ (x : Thing), Person x → ∃ y, Person y ∧ Knows x yx:ThinghXPerson:Person xhIfXPersonThenKnowsSomeone:Person x → ∃ y, Person y ∧ Knows x yhXKnowsSomeone:∃ y, Person y ∧ Knows x yy:ThinghYPersonAndXKnowsY:Person y ∧ Knows x y⊢ Person y exact hYPersonAndXKnowsY.leftAll goals completed! 🐙

Existential quantification and negation

A witness submitted but received no confirmation. The universal rule produces a confirmation for that witness, and the negation produces False.

-- Combine ∃E, ∧E, ∀E, and ¬E.
theorem lecture05_exists_negation_combined
    (Thing : Type)
    (Submitted ReceivedConfirmation : Thing → Prop)
    (hSubmissionGetsConfirmation :
      ∀ x : Thing, Submitted x → ReceivedConfirmation x)
    (hCounterexample :
      ∃ x : Thing, Submitted x ∧ ¬ReceivedConfirmation x) :
    False := byThing:TypeSubmitted:Thing → PropReceivedConfirmation:Thing → ProphSubmissionGetsConfirmation:∀ (x : Thing), Submitted x → ReceivedConfirmation xhCounterexample:∃ x, Submitted x ∧ ¬ReceivedConfirmation x⊢ False
  apply Exists.elim hCounterexampleThing:TypeSubmitted:Thing → PropReceivedConfirmation:Thing → ProphSubmissionGetsConfirmation:∀ (x : Thing), Submitted x → ReceivedConfirmation xhCounterexample:∃ x, Submitted x ∧ ¬ReceivedConfirmation x⊢ ∀ (a : Thing), Submitted a ∧ ¬ReceivedConfirmation a → False
  intro yThing:TypeSubmitted:Thing → PropReceivedConfirmation:Thing → ProphSubmissionGetsConfirmation:∀ (x : Thing), Submitted x → ReceivedConfirmation xhCounterexample:∃ x, Submitted x ∧ ¬ReceivedConfirmation xy:Thing⊢ Submitted y ∧ ¬ReceivedConfirmation y → False
  intro hYSubmittedAndNoConfirmationThing:TypeSubmitted:Thing → PropReceivedConfirmation:Thing → ProphSubmissionGetsConfirmation:∀ (x : Thing), Submitted x → ReceivedConfirmation xhCounterexample:∃ x, Submitted x ∧ ¬ReceivedConfirmation xy:ThinghYSubmittedAndNoConfirmation:Submitted y ∧ ¬ReceivedConfirmation y⊢ False
  have hYSubmitted := hYSubmittedAndNoConfirmation.leftThing:TypeSubmitted:Thing → PropReceivedConfirmation:Thing → ProphSubmissionGetsConfirmation:∀ (x : Thing), Submitted x → ReceivedConfirmation xhCounterexample:∃ x, Submitted x ∧ ¬ReceivedConfirmation xy:ThinghYSubmittedAndNoConfirmation:Submitted y ∧ ¬ReceivedConfirmation yhYSubmitted:Submitted y⊢ False
  have hYNoConfirmation := hYSubmittedAndNoConfirmation.rightThing:TypeSubmitted:Thing → PropReceivedConfirmation:Thing → ProphSubmissionGetsConfirmation:∀ (x : Thing), Submitted x → ReceivedConfirmation xhCounterexample:∃ x, Submitted x ∧ ¬ReceivedConfirmation xy:ThinghYSubmittedAndNoConfirmation:Submitted y ∧ ¬ReceivedConfirmation yhYSubmitted:Submitted yhYNoConfirmation:¬ReceivedConfirmation y⊢ False
  have hIfYSubmittedThenConfirmation := hSubmissionGetsConfirmation yThing:TypeSubmitted:Thing → PropReceivedConfirmation:Thing → ProphSubmissionGetsConfirmation:∀ (x : Thing), Submitted x → ReceivedConfirmation xhCounterexample:∃ x, Submitted x ∧ ¬ReceivedConfirmation xy:ThinghYSubmittedAndNoConfirmation:Submitted y ∧ ¬ReceivedConfirmation yhYSubmitted:Submitted yhYNoConfirmation:¬ReceivedConfirmation yhIfYSubmittedThenConfirmation:Submitted y → ReceivedConfirmation y⊢ False
  have hYReceivedConfirmation := hIfYSubmittedThenConfirmation hYSubmittedThing:TypeSubmitted:Thing → PropReceivedConfirmation:Thing → ProphSubmissionGetsConfirmation:∀ (x : Thing), Submitted x → ReceivedConfirmation xhCounterexample:∃ x, Submitted x ∧ ¬ReceivedConfirmation xy:ThinghYSubmittedAndNoConfirmation:Submitted y ∧ ¬ReceivedConfirmation yhYSubmitted:Submitted yhYNoConfirmation:¬ReceivedConfirmation yhIfYSubmittedThenConfirmation:Submitted y → ReceivedConfirmation yhYReceivedConfirmation:ReceivedConfirmation y⊢ False
  have hContradiction := hYNoConfirmation hYReceivedConfirmationThing:TypeSubmitted:Thing → PropReceivedConfirmation:Thing → ProphSubmissionGetsConfirmation:∀ (x : Thing), Submitted x → ReceivedConfirmation xhCounterexample:∃ x, Submitted x ∧ ¬ReceivedConfirmation xy:ThinghYSubmittedAndNoConfirmation:Submitted y ∧ ¬ReceivedConfirmation yhYSubmitted:Submitted yhYNoConfirmation:¬ReceivedConfirmation yhIfYSubmittedThenConfirmation:Submitted y → ReceivedConfirmation yhYReceivedConfirmation:ReceivedConfirmation yhContradiction:False⊢ False
  exact hContradictionAll goals completed! 🐙

Existential quantification and biconditional

We open the existential, instantiate the biconditional at its witness, and use Iff.mp to extract the required direction. The same object witnesses the goal.

-- Combine ∃E, ∀E, ↔E, and ∃I.
-- Being a square is equivalent to having four equal sides. A square exists;
-- therefore, something has four equal sides.
theorem lecture05_exists_iff_combined
    (Thing : Type)
    (Square HasFourEqualSides : Thing → Prop)
    (hEquivalence :
      ∀ x : Thing, Square x ↔ HasFourEqualSides x)
    (hSquareExists : ∃ x : Thing, Square x) :
    ∃ x : Thing, HasFourEqualSides x := byThing:TypeSquare:Thing → PropHasFourEqualSides:Thing → ProphEquivalence:∀ (x : Thing), Square x ↔ HasFourEqualSides xhSquareExists:∃ x, Square x⊢ ∃ x, HasFourEqualSides x
  apply Exists.elim hSquareExistsThing:TypeSquare:Thing → PropHasFourEqualSides:Thing → ProphEquivalence:∀ (x : Thing), Square x ↔ HasFourEqualSides xhSquareExists:∃ x, Square x⊢ ∀ (a : Thing), Square a → ∃ x, HasFourEqualSides x
  intro yThing:TypeSquare:Thing → PropHasFourEqualSides:Thing → ProphEquivalence:∀ (x : Thing), Square x ↔ HasFourEqualSides xhSquareExists:∃ x, Square xy:Thing⊢ Square y → ∃ x, HasFourEqualSides x
  intro hYSquareThing:TypeSquare:Thing → PropHasFourEqualSides:Thing → ProphEquivalence:∀ (x : Thing), Square x ↔ HasFourEqualSides xhSquareExists:∃ x, Square xy:ThinghYSquare:Square y⊢ ∃ x, HasFourEqualSides x
  apply Exists.intro yThing:TypeSquare:Thing → PropHasFourEqualSides:Thing → ProphEquivalence:∀ (x : Thing), Square x ↔ HasFourEqualSides xhSquareExists:∃ x, Square xy:ThinghYSquare:Square y⊢ HasFourEqualSides y
  have hEquivalenceForY := hEquivalence yThing:TypeSquare:Thing → PropHasFourEqualSides:Thing → ProphEquivalence:∀ (x : Thing), Square x ↔ HasFourEqualSides xhSquareExists:∃ x, Square xy:ThinghYSquare:Square yhEquivalenceForY:Square y ↔ HasFourEqualSides y⊢ HasFourEqualSides y
  have hForwardDirection := Iff.mp hEquivalenceForYThing:TypeSquare:Thing → PropHasFourEqualSides:Thing → ProphEquivalence:∀ (x : Thing), Square x ↔ HasFourEqualSides xhSquareExists:∃ x, Square xy:ThinghYSquare:Square yhEquivalenceForY:Square y ↔ HasFourEqualSides yhForwardDirection:Square y → HasFourEqualSides y⊢ HasFourEqualSides y
  have hYHasFourEqualSides := hForwardDirection hYSquareThing:TypeSquare:Thing → PropHasFourEqualSides:Thing → ProphEquivalence:∀ (x : Thing), Square x ↔ HasFourEqualSides xhSquareExists:∃ x, Square xy:ThinghYSquare:Square yhEquivalenceForY:Square y ↔ HasFourEqualSides yhForwardDirection:Square y → HasFourEqualSides yhYHasFourEqualSides:HasFourEqualSides y⊢ HasFourEqualSides y
  exact hYHasFourEqualSidesAll goals completed! 🐙

In each example, we first handle the outer constructor of the goal, make local witnesses explicit, and then apply the propositional rules.

9. Working strategy

When reading a quantified formula, ask:

Does the goal begin with ∀? Introduce an arbitrary object with intro.
Is there a universal assumption? Apply it to a term from the domain.
Does the goal begin with ∃? Choose a witness with Exists.intro.
Is there an existential assumption? Open it with Exists.elim without presupposing the witness's identity.
Are several quantifiers nested? Check their order and dependencies.

This strategy extends the one used for propositional connectives. Inspect the main constructor of the goal for an introduction rule and compound assumptions for useful elimination rules.

10. Sources and further reading

Jeremy Avigad, Robert Y. Lewis, and Floris van Doorn, Logic and Proof, “Natural Deduction Rules”: https://leanprover.github.io/logic_and_proof_lean3/nd_quickref.html.
Lean developers, The Lean Language Reference, “Quantifiers”: https://lean-lang.org/doc/reference/latest/Basic-Propositions/Quantifiers/.
Jeremy Avigad, Robert Y. Lewis, and Floris van Doorn, Logic and Proof, “First Order Logic in Lean”: https://leanprover.github.io/logic_and_proof_lean3/first_order_logic_in_lean.html.
Jeremy Avigad, Robert Y. Lewis, and Floris van Doorn, Logic and Proof, “Natural Deduction for First Order Logic”: https://leanprover.github.io/logic_and_proof_lean3/natural_deduction_for_first_order_logic.html.
Jeremy Avigad, Leonardo de Moura, Soonho Kong, and Sebastian Ullrich, Theorem Proving in Lean 4, “Classical Logic”: https://lean-lang.org/theorem_proving_in_lean4/Propositions-and-Proofs/.