This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem bj-pm11.53a

Description: A variant of pm11.53v . One can similarly prove a variant with DV ( y , ph ) and A. y F// x ps instead of DV ( x , ps ) and A. x F// y ph . (Contributed by BJ, 7-Oct-2024)

Ref Expression
Assertion bj-pm11.53a ⊢ ∀ x Ⅎ' y φ → ∀ x ∀ y φ → ψ ↔ ∃ x φ → ∀ y ψ

Proof

Step Hyp Ref Expression
1 bj-nnfv ⊢ Ⅎ' x ∀ y ψ
2 bj-pm11.53vw ⊢ ∀ x Ⅎ' y φ ∧ Ⅎ' x ∀ y ψ → ∀ x ∀ y φ → ψ ↔ ∃ x φ → ∀ y ψ
3 1 2 mpan2 ⊢ ∀ x Ⅎ' y φ → ∀ x ∀ y φ → ψ ↔ ∃ x φ → ∀ y ψ