Theorem cbvrexfw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexf with a disjoint variable condition, which does not require ax-13 . For a version not dependent on ax-11 and ax-12, see cbvrexvw . (Contributed by FL, 27-Apr-2008) Avoid ax-10 , ax-13 . (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvrexfw.1	|- F/_ x A
	cbvrexfw.2	|- F/_ y A
	cbvrexfw.3	|- F/ y ph
	cbvrexfw.4	|- F/ x ps
	cbvrexfw.5	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbvrexfw	|- ( E. x e. A ph <-> E. y e. A ps )

Proof

Step	Hyp	Ref	Expression
1		cbvrexfw.1	|- F/_ x A
2		cbvrexfw.2	|- F/_ y A
3		cbvrexfw.3	|- F/ y ph
4		cbvrexfw.4	|- F/ x ps
5		cbvrexfw.5	|- ( x = y -> ( ph <-> ps ) )
6	3	nfn	|- F/ y -. ph
7	4	nfn	|- F/ x -. ps
8	5	notbid	|- ( x = y -> ( -. ph <-> -. ps ) )
9	1 2 6 7 8	cbvralfw	|- ( A. x e. A -. ph <-> A. y e. A -. ps )
10		ralnex	|- ( A. x e. A -. ph <-> -. E. x e. A ph )
11		ralnex	|- ( A. y e. A -. ps <-> -. E. y e. A ps )
12	9 10 11	3bitr3i	|- ( -. E. x e. A ph <-> -. E. y e. A ps )
13	12	con4bii	|- ( E. x e. A ph <-> E. y e. A ps )