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Metamath Proof Explorer

Theorem sbcgfi

Description: Substitution for a variable not free in a wff does not affect it, in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019)

	Ref	Expression
Hypotheses	sbcgfi.1	$⊢ A \in V$
	sbcgfi.2	$⊢ Ⅎ x φ$
Assertion	sbcgfi	$⊢ \underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow φ$

Proof

Step	Hyp	Ref	Expression
1		sbcgfi.1	$⊢ A \in V$
2		sbcgfi.2	$⊢ Ⅎ x φ$
3	2	sbcgf	$⊢ A \in V \to (\underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow φ)$
4	1 3	ax-mp	$⊢ \underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow φ$