Theorem cbvalv1

Description: Rule used to change bound variables, using implicit substitution. Version of cbval with a disjoint variable condition, which does not require ax-13 . See cbvalvw for a version with two more disjoint variable conditions, requiring fewer axioms, and cbvalv for another variant. (Contributed by NM, 13-May-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	cbvalv1	|- ( A. x ph <-> A. 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	3	biimpd	|- ( x = y -> ( ph -> ps ) )
5	1 2 4	cbv3v	|- ( A. x ph -> A. y ps )
6	3	biimprd	|- ( x = y -> ( ps -> ph ) )
7	6	equcoms	|- ( y = x -> ( ps -> ph ) )
8	2 1 7	cbv3v	|- ( A. y ps -> A. x ph )
9	5 8	impbii	|- ( A. x ph <-> A. y ps )