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	$$⊢A\in \mathrm{V}$$
	sbcgfi.2	$$⊢Ⅎx\phi$$
Assertion	sbcgfi	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \phi$$

Proof

Step	Hyp	Ref	Expression
1		sbcgfi.1	$$⊢A\in \mathrm{V}$$
2		sbcgfi.2	$$⊢Ⅎx\phi$$
3	2	sbcgf	$$⊢A\in \mathrm{V}\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \phi \right)$$
4	1 3	ax-mp	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \phi$$