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	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		nfv	$$⊢Ⅎy\phi$$
2		nfs1v	$$⊢Ⅎx\left[y/x\right]\phi$$
3		sbequ12	$$⊢x=y\to \left(\phi \leftrightarrow \left[y/x\right]\phi \right)$$
4	1 2 3	cbvrexw	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\left[y/x\right]\phi$$