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Metamath Proof Explorer

Theorem pm11.53v

Description: Version of pm11.53 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020)

Ref Expression
Assertion pm11.53v ⊢ ∀ x ∀ y φ → ψ ↔ ∃ x φ → ∀ y ψ

Proof

Step Hyp Ref Expression
1 19.21v ⊢ ∀ y φ → ψ ↔ φ → ∀ y ψ
2 1 albii ⊢ ∀ x ∀ y φ → ψ ↔ ∀ x φ → ∀ y ψ
3 19.23v ⊢ ∀ x φ → ∀ y ψ ↔ ∃ x φ → ∀ y ψ
4 2 3 bitri ⊢ ∀ x ∀ y φ → ψ ↔ ∃ x φ → ∀ y ψ