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Metamath Proof Explorer

Theorem sbcgf

Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004) (Proof shortened by Andrew Salmon, 29-Jun-2011)

		Ref	Expression
	Hypothesis	sbcgf.1	$⊢ Ⅎ x φ$
	Assertion	sbcgf	$⊢ A \in V \to (\underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow φ)$

Proof

Step	Hyp	Ref	Expression
1		sbcgf.1	$⊢ Ⅎ x φ$
2		sbctt	$⊢ (A \in V \land Ⅎ x φ) \to (\underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow φ)$
3	1 2	mpan2	$⊢ A \in V \to (\underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow φ)$