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Metamath Proof Explorer

Theorem sb2ae

Description: In the case of two successive substitutions for two always equal variables, the second substitution has no effect. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by BJ and WL, 9-Aug-2023) (New usage is discouraged.)

		Ref	Expression
	Assertion	sb2ae	$⊢ \forall x x = y \to ([u/ x] [v/ y] φ \leftrightarrow [v/ y] φ)$

Proof

Step	Hyp	Ref	Expression
1		drsb1	$⊢ \forall x x = y \to ([u/ x] [v/ y] φ \leftrightarrow [u/ y] [v/ y] φ)$
2		nfs1v	$⊢ Ⅎ y [v/ y] φ$
3	2	sbf	$⊢ [u/ y] [v/ y] φ \leftrightarrow [v/ y] φ$
4	1 3	bitrdi	$⊢ \forall x x = y \to ([u/ x] [v/ y] φ \leftrightarrow [v/ y] φ)$