Theorem bj-cbvexdv

Description: Version of cbvexd 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-cbvexdv	|- ( ph -> ( E. x ps <-> E. 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	2	nfnd	|- ( ph -> F/ y -. ps )
5		notbi	|- ( ( ps <-> ch ) <-> ( -. ps <-> -. ch ) )
6	3 5	syl6ib	|- ( ph -> ( x = y -> ( -. ps <-> -. ch ) ) )
7	1 4 6	bj-cbvaldv	|- ( ph -> ( A. x -. ps <-> A. y -. ch ) )
8	7	notbid	|- ( ph -> ( -. A. x -. ps <-> -. A. y -. ch ) )
9		df-ex	|- ( E. x ps <-> -. A. x -. ps )
10		df-ex	|- ( E. y ch <-> -. A. y -. ch )
11	8 9 10	3bitr4g	|- ( ph -> ( E. x ps <-> E. y ch ) )