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Metamath Proof Explorer

Theorem sbco2v

Description: A composition law for substitution. Version of sbco2 with disjoint variable conditions but not requiring ax-13 . (Contributed by NM, 30-Jun-1994) (Revised by Wolf Lammen, 29-Apr-2023)

		Ref	Expression
	Hypothesis	sbco2v.1	$⊢ Ⅎ z φ$
	Assertion	sbco2v	$⊢ [y/ z] [z/ x] φ \leftrightarrow [y/ x] φ$

Proof

Step	Hyp	Ref	Expression
1		sbco2v.1	$⊢ Ⅎ z φ$
2	1	nfsbv	$⊢ Ⅎ z [y/ x] φ$
3		sbequ	$⊢ z = y \to ([z/ x] φ \leftrightarrow [y/ x] φ)$
4	2 3	sbiev	$⊢ [y/ z] [z/ x] φ \leftrightarrow [y/ x] φ$