Theorem cbvex2v

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2 with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 14-Sep-2003) (Revised by BJ, 16-Jun-2019)

	Ref	Expression
Hypotheses	cbval2v.1	|- F/ z ph
	cbval2v.2	|- F/ w ph
	cbval2v.3	|- F/ x ps
	cbval2v.4	|- F/ y ps
	cbval2v.5	|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
Assertion	cbvex2v	|- ( E. x E. y ph <-> E. z E. w ps )

Proof

Step	Hyp	Ref	Expression
1		cbval2v.1	|- F/ z ph
2		cbval2v.2	|- F/ w ph
3		cbval2v.3	|- F/ x ps
4		cbval2v.4	|- F/ y ps
5		cbval2v.5	|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
6	1	nfn	|- F/ z -. ph
7	2	nfn	|- F/ w -. ph
8	3	nfn	|- F/ x -. ps
9	4	nfn	|- F/ y -. ps
10	5	notbid	|- ( ( x = z /\ y = w ) -> ( -. ph <-> -. ps ) )
11	6 7 8 9 10	cbval2v	|- ( A. x A. y -. ph <-> A. z A. w -. ps )
12		2nexaln	|- ( -. E. x E. y ph <-> A. x A. y -. ph )
13		2nexaln	|- ( -. E. z E. w ps <-> A. z A. w -. ps )
14	11 12 13	3bitr4i	|- ( -. E. x E. y ph <-> -. E. z E. w ps )
15	14	con4bii	|- ( E. x E. y ph <-> E. z E. w ps )