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Metamath Proof Explorer

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 GG, 10-Jan-2024) (Proof shortened by Wolf Lammen, 8-Mar-2025)

		Ref	Expression
	Assertion	cbvrexsvw	$⊢ \exists x \in A φ \leftrightarrow \exists y \in A [y/ x] φ$

Proof

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