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Metamath Proof Explorer

Theorem sbco2vv

Description: A composition law for substitution. Version of sbco2 with disjoint variable conditions and fewer axioms. (Contributed by NM, 30-Jun-1994) (Revised by BJ, 22-Dec-2020) (Proof shortened by Wolf Lammen, 29-Apr-2023)

		Ref	Expression
	Assertion	sbco2vv	$⊢ [y/ z] [z/ x] φ \leftrightarrow [y/ x] φ$

Proof

Step	Hyp	Ref	Expression
1		sbequ	$⊢ z = w \to ([z/ x] φ \leftrightarrow [w/ x] φ)$
2		sbequ	$⊢ w = y \to ([w/ x] φ \leftrightarrow [y/ x] φ)$
3	1 2	sbievw2	$⊢ [y/ z] [z/ x] φ \leftrightarrow [y/ x] φ$