Theorem cbvralw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvral with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 31-Jul-2003) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvralw.1	|- F/ y ph
	cbvralw.2	|- F/ x ps
	cbvralw.3	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbvralw	|- ( A. x e. A ph <-> A. y e. A ps )

Proof

Step	Hyp	Ref	Expression
1		cbvralw.1	|- F/ y ph
2		cbvralw.2	|- F/ x ps
3		cbvralw.3	|- ( x = y -> ( ph <-> ps ) )
4		nfcv	|- F/_ x A
5		nfcv	|- F/_ y A
6	4 5 1 2 3	cbvralfw	|- ( A. x e. A ph <-> A. y e. A ps )