Theorem bj-nnf-cbvali

Description: Compared with bj-nnf-cbvaliv , replacing the DV condition on y , ps with the nonfreeness condition requires ax-11 . (Contributed by BJ, 4-Apr-2026)

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

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-cbvali.nf0	|- ( ph -> A. x ph )
2		bj-nnf-cbvali.nf1	|- ( ph -> A. y ph )
3		bj-nnf-cbvali.ps	|- ( ph -> F// y ps )
4		bj-nnf-cbvali.ch	|- ( ph -> F// x ch )
5		bj-nnf-cbvali.is	|- ( ( ph /\ x = y ) -> ( ps -> ch ) )
6	3	bj-nnfad	|- ( ph -> ( ps -> A. y ps ) )
7	1 6	hbald	|- ( ph -> ( A. x ps -> A. y A. x ps ) )
8	1 4 5	bj-nnf-spim	|- ( ph -> ( A. x ps -> ch ) )
9	2 7 8	bj-alrimd	|- ( ph -> ( A. x ps -> A. y ch ) )