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	$$⊢\phi \leftrightarrow \psi$$
Assertion	bj-nnfbii	$$⊢Ⅎ\text{'}x\phi \leftrightarrow Ⅎ\text{'}x\psi$$

Proof

Step	Hyp	Ref	Expression
1		bj-nnfbii.1	$$⊢\phi \leftrightarrow \psi$$
2		bj-nnfbi	$$⊢\left(\left(\phi \leftrightarrow \psi \right)\wedge \forall x\left(\phi \leftrightarrow \psi \right)\right)\to \left(Ⅎ\text{'}x\phi \leftrightarrow Ⅎ\text{'}x\psi \right)$$
3	1 2	bj-mpgs	$$⊢Ⅎ\text{'}x\phi \leftrightarrow Ⅎ\text{'}x\psi$$