Theorem cbvexv1

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex with a disjoint variable condition, which does not require ax-13 . See cbvexvw for a version with two disjoint variable conditions, requiring fewer axioms, and cbvexv for another variant. (Contributed by NM, 21-Jun-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	cbvalv1.nf1	|- F/ y ph
	cbvalv1.nf2	|- F/ x ps
	cbvalv1.1	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbvexv1	|- ( E. x ph <-> E. y ps )

Proof

Step	Hyp	Ref	Expression
1		cbvalv1.nf1	|- F/ y ph
2		cbvalv1.nf2	|- F/ x ps
3		cbvalv1.1	|- ( 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	cbvalv1	|- ( 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 )