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) Avoid ax-13 . (Revised by Gino Giotto, 10-Jan-2024) (Proof shortened by Wolf Lammen, 8-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		nfv	|- F/ y ph
2		nfs1v	|- F/ x [ y / x ] ph
3		sbequ12	|- ( x = y -> ( ph <-> [ y / x ] ph ) )
4	1 2 3	cbvrexw	|- ( E. x e. A ph <-> E. y e. A [ y / x ] ph )