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Metamath Proof Explorer

Theorem cbvralsvwOLD

Description: Obsolete version of cbvralsvw as of 21-Aug-2025. (Contributed by NM, 20-Nov-2005) Avoid ax-13 . (Revised by GG, 10-Jan-2024) (Proof shortened by Wolf Lammen, 8-Mar-2025) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	cbvralsvwOLD	$⊢ \forall x \in A φ \leftrightarrow \forall y \in A [y/ x] φ$

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	cbvralw	$⊢ \forall x \in A φ \leftrightarrow \forall y \in A [y/ x] φ$