Theorem cbvralsvwOLDOLD

Description: Obsolete version of cbvralsvw as of 8-Mar-2025. (Contributed by NM, 20-Nov-2005) Avoid ax-13 . (Revised by GG, 10-Jan-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	cbvralsvwOLDOLD	|- ( 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 )