Theorem cbvex

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Check out cbvexvw , cbvexv1 for weaker versions requiring fewer axioms. (Contributed by NM, 21-Jun-1993) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbval.1	|- F/ y ph
	cbval.2	|- F/ x ps
	cbval.3	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbvex	|- ( E. x ph <-> E. y ps )

Proof

Step	Hyp	Ref	Expression
1		cbval.1	|- F/ y ph
2		cbval.2	|- F/ x ps
3		cbval.3	|- ( x = y -> ( ph <-> ps ) )
4	1	nfn	|- F/ y -. ph
5	2	nfn	|- F/ x -. ps
6	3	notbid	|- ( x = y -> ( -. ph <-> -. ps ) )
7	4 5 6	cbval	|- ( A. x -. ph <-> A. y -. ps )
8		alnex	|- ( A. x -. ph <-> -. E. x ph )
9		alnex	|- ( A. y -. ps <-> -. E. y ps )
10	7 8 9	3bitr3i	|- ( -. E. x ph <-> -. E. y ps )
11	10	con4bii	|- ( E. x ph <-> E. y ps )