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	|- ( ph -> ( ps <-> ch ) )
Assertion	bj-nnfbd	|- ( ph -> ( F// x ps <-> F// x ch ) )

Proof

Step	Hyp	Ref	Expression
1		bj-nnfbd.1	|- ( ph -> ( ps <-> ch ) )
2	1	alrimiv	|- ( ph -> A. x ( ps <-> ch ) )
3		bj-nnfbi	|- ( ( ( ps <-> ch ) /\ A. x ( ps <-> ch ) ) -> ( F// x ps <-> F// x ch ) )
4	1 2 3	syl2anc	|- ( ph -> ( F// x ps <-> F// x ch ) )