Theorem sbciegf

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005) (Revised by Mario Carneiro, 13-Oct-2016)

	Ref	Expression
Hypotheses	sbciegf.1	$$⊢Ⅎx\psi$$
	sbciegf.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	sbciegf	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbciegf.1	$$⊢Ⅎx\psi$$
2		sbciegf.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3	2	ax-gen	$$⊢\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)$$
4		sbciegft	$$⊢\left(A\in V\wedge Ⅎx\psi \wedge \forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \psi \right)$$
5	1 3 4	mp3an23	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \psi \right)$$