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 ( 𝜑𝜓 )
Assertion bj-nnfbii ( Ⅎ' 𝑥 𝜑 ↔ Ⅎ' 𝑥 𝜓 )

Proof

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