Theorem cbvralsvw

Description: Change bound variable by using a substitution. Version of cbvralsv with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 20-Nov-2005) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Assertion	cbvralsvw	|- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )

Proof

Step	Hyp	Ref	Expression
1		nfv	|- F/ z ph
2		nfs1v	|- F/ x [ z / x ] ph
3		sbequ12	|- ( x = z -> ( ph <-> [ z / x ] ph ) )
4	1 2 3	cbvralw	|- ( A. x e. A ph <-> A. z e. A [ z / x ] ph )
5		nfv	|- F/ y [ z / x ] ph
6		nfv	|- F/ z [ y / x ] ph
7		sbequ	|- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
8	5 6 7	cbvralw	|- ( A. z e. A [ z / x ] ph <-> A. y e. A [ y / x ] ph )
9	4 8	bitri	|- ( A. x e. A ph <-> A. y e. A [ y / x ] ph )