Theorem cbvrexsvw

Description: Change bound variable by using a substitution. Version of cbvrexsv with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 2-Mar-2008) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Assertion	cbvrexsvw	|- ( E. x e. A ph <-> E. 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	cbvrexw	|- ( E. x e. A ph <-> E. 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	cbvrexw	|- ( E. z e. A [ z / x ] ph <-> E. y e. A [ y / x ] ph )
9	4 8	bitri	|- ( E. x e. A ph <-> E. y e. A [ y / x ] ph )