2.5. The Order of Quantifiers🔗

Consecutive quantifiers of the same kind commute, and quantifiers of different kinds do not. One direction of the exchange holds. A witness that satisfies R with every y in particular satisfies R with each given y.

theorem exists_forall_swap (α β : Type) (R : α → β → Prop)
    (h : ∃ x, ∀ y, R x y) : ∀ y, ∃ x, R x y := byα:Typeβ:TypeR:α → β → Proph:∃ x, ∀ (y : β), R x y⊢ ∀ (y : β), ∃ x, R x y
  intro bα:Typeβ:TypeR:α → β → Proph:∃ x, ∀ (y : β), R x yb:β⊢ ∃ x, R x b
  obtain ⟨a, ha⟩ := hα:Typeβ:TypeR:α → β → Propb:βa:αha:∀ (y : β), R a y⊢ ∃ x, R x b
  exact ⟨a, ha b⟩All goals completed! 🐙

The converse fails. Over the natural numbers, take R x y to be x ≥ y. Then ∀ y, ∃ x, R x y holds, since each y satisfies y ≥ y, and ∃ x, ∀ y, R x y states that some natural number is greater than or equal to every natural number, which is false.