Theorem bj-cbvaldv

Description: Version of cbvald with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbvaldv.1	|- F/ y ph
	bj-cbvaldv.2	|- ( ph -> F/ y ps )
	bj-cbvaldv.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )
Assertion	bj-cbvaldv	|- ( ph -> ( A. x ps <-> A. y ch ) )

Proof

Step	Hyp	Ref	Expression
1		bj-cbvaldv.1	|- F/ y ph
2		bj-cbvaldv.2	|- ( ph -> F/ y ps )
3		bj-cbvaldv.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )
4		nfv	|- F/ x ph
5		nfv	|- F/ x ch
6	5	a1i	|- ( ph -> F/ x ch )
7	4 1 2 6 3	bj-cbv2v	|- ( ph -> ( A. x ps <-> A. y ch ) )