Theorem bj-nnfbd

Description: If two formulas are equivalent for all x , then nonfreeness of x in one of them is equivalent to nonfreeness in the other, deduction form. See bj-nnfbi . (Contributed by BJ, 27-Aug-2023)

	Ref	Expression
Hypothesis	bj-nnfbd.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	bj-nnfbd	$$⊢\phi \to \left(Ⅎ\text{'}x\psi \leftrightarrow Ⅎ\text{'}x\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-nnfbd.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	alrimiv	$$⊢\phi \to \forall x\left(\psi \leftrightarrow \chi \right)$$
3		bj-nnfbi	$$⊢\left(\left(\psi \leftrightarrow \chi \right)\wedge \forall x\left(\psi \leftrightarrow \chi \right)\right)\to \left(Ⅎ\text{'}x\psi \leftrightarrow Ⅎ\text{'}x\chi \right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(Ⅎ\text{'}x\psi \leftrightarrow Ⅎ\text{'}x\chi \right)$$