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Metamath Proof Explorer

Theorem pm11.53

Description: Theorem *11.53 in WhiteheadRussell p. 164. See pm11.53v for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011)

Ref Expression
Assertion pm11.53 ⊢ ∀ 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 nfv ⊢ Ⅎ x ψ
4 3 nfal ⊢ Ⅎ x ∀ y ψ
5 4 19.23 ⊢ ∀ x φ → ∀ y ψ ↔ ∃ x φ → ∀ y ψ
6 2 5 bitri ⊢ ∀ x ∀ y φ → ψ ↔ ∃ x φ → ∀ y ψ