Theorem cbvex2

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvex2v if possible. (Contributed by NM, 14-Sep-2003) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 16-Jun-2019) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbval2.1	|- F/ z ph
	cbval2.2	|- F/ w ph
	cbval2.3	|- F/ x ps
	cbval2.4	|- F/ y ps
	cbval2.5	|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
Assertion	cbvex2	|- ( E. x E. y ph <-> E. z E. w ps )

Proof

Step	Hyp	Ref	Expression
1		cbval2.1	|- F/ z ph
2		cbval2.2	|- F/ w ph
3		cbval2.3	|- F/ x ps
4		cbval2.4	|- F/ y ps
5		cbval2.5	|- ( ( 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	cbval2	|- ( 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 )