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Metamath Proof Explorer

Theorem bj-nnfbii

Description: If two formulas are equivalent for all x , then nonfreeness of x in one of them is equivalent to nonfreeness in the other, inference form. See bj-nnfbi . (Contributed by BJ, 18-Nov-2023)

Ref Expression
Hypothesis bj-nnfbii.1 ⊢ φ ↔ ψ
Assertion bj-nnfbii ⊢ Ⅎ' x φ ↔ Ⅎ' x ψ

Proof

Step Hyp Ref Expression
1 bj-nnfbii.1 ⊢ φ ↔ ψ
2 bj-nnfbi ⊢ φ ↔ ψ ∧ ∀ x φ ↔ ψ → Ⅎ' x φ ↔ Ⅎ' x ψ
3 1 2 bj-mpgs ⊢ Ⅎ' x φ ↔ Ⅎ' x ψ