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Metamath Proof Explorer

Theorem bj-pm11.53v

Description: Version of pm11.53v with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024)

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

Proof

Step Hyp Ref Expression
1 bj-nnfalt ⊢ ∀ y Ⅎ' x ψ → 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 sylan2 ⊢ ∀ x Ⅎ' y φ ∧ ∀ y Ⅎ' x ψ → ∀ x ∀ y φ → ψ ↔ ∃ x φ → ∀ y ψ