Theorem sbcied2

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014)

	Ref	Expression
Hypotheses	sbcied2.1	$$⊢\phi \to A\in V$$
	sbcied2.2	$$⊢\phi \to A=B$$
	sbcied2.3	$$⊢\left(\phi \wedge x=B\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	sbcied2	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbcied2.1	$$⊢\phi \to A\in V$$
2		sbcied2.2	$$⊢\phi \to A=B$$
3		sbcied2.3	$$⊢\left(\phi \wedge x=B\right)\to \left(\psi \leftrightarrow \chi \right)$$
4		id	$$⊢x=A\to x=A$$
5	4 2	sylan9eqr	$$⊢\left(\phi \wedge x=A\right)\to x=B$$
6	5 3	syldan	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
7	1 6	sbcied	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \chi \right)$$