Theorem sbcie

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004)

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

Proof

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