Theorem bj-nnf-cbval

Description: Compared with cbvalv1 , this saves ax-12 . (Contributed by BJ, 4-Apr-2026)

	Ref	Expression
Hypotheses	bj-nnf-cbval.nf0	|- ( ph -> A. x ph )
	bj-nnf-cbval.nf1	|- ( ph -> A. y ph )
	bj-nnf-cbval.ps	|- ( ph -> F// y ps )
	bj-nnf-cbval.ch	|- ( ph -> F// x ch )
	bj-nnf-cbval.is	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
Assertion	bj-nnf-cbval	|- ( ph -> ( A. x ps <-> A. y ch ) )

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-cbval.nf0	|- ( ph -> A. x ph )
2		bj-nnf-cbval.nf1	|- ( ph -> A. y ph )
3		bj-nnf-cbval.ps	|- ( ph -> F// y ps )
4		bj-nnf-cbval.ch	|- ( ph -> F// x ch )
5		bj-nnf-cbval.is	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
6	5	biimpd	|- ( ( ph /\ x = y ) -> ( ps -> ch ) )
7	1 2 3 4 6	bj-nnf-cbvali	|- ( ph -> ( A. x ps -> A. y ch ) )
8		equcomi	|- ( y = x -> x = y )
9	8 5	sylan2	|- ( ( ph /\ y = x ) -> ( ps <-> ch ) )
10	9	biimprd	|- ( ( ph /\ y = x ) -> ( ch -> ps ) )
11	2 1 4 3 10	bj-nnf-cbvali	|- ( ph -> ( A. y ch -> A. x ps ) )
12	7 11	impbid	|- ( ph -> ( A. x ps <-> A. y ch ) )