Theorem cbv3v

Description: Rule used to change bound variables, using implicit substitution. Version of cbv3 with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	cbv3v.nf1	|- F/ y ph
	cbv3v.nf2	|- F/ x ps
	cbv3v.1	|- ( x = y -> ( ph -> ps ) )
Assertion	cbv3v	|- ( A. x ph -> A. y ps )

Proof

Step	Hyp	Ref	Expression
1		cbv3v.nf1	|- F/ y ph
2		cbv3v.nf2	|- F/ x ps
3		cbv3v.1	|- ( x = y -> ( ph -> ps ) )
4	1	nf5ri	|- ( ph -> A. y ph )
5	4	hbal	|- ( A. x ph -> A. y A. x ph )
6	2 3	spimfv	|- ( A. x ph -> ps )
7	5 6	alrimih	|- ( A. x ph -> A. y ps )