Theorem cbvrexf

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 cbvrexfw when possible. (Contributed by FL, 27-Apr-2008) (Revised by Mario Carneiro, 9-Oct-2016) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvralf.1	|- F/_ x A
2		cbvralf.2	|- F/_ y A
3		cbvralf.3	|- F/ y ph
4		cbvralf.4	|- F/ x ps
5		cbvralf.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	cbvralf	|- ( A. x e. A -. ph <-> A. y e. A -. ps )
10	9	notbii	|- ( -. A. x e. A -. ph <-> -. A. y e. A -. ps )
11		dfrex2	|- ( E. x e. A ph <-> -. A. x e. A -. ph )
12		dfrex2	|- ( E. y e. A ps <-> -. A. y e. A -. ps )
13	10 11 12	3bitr4i	|- ( E. x e. A ph <-> E. y e. A ps )