Theorem sbciedf

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014)

	Ref	Expression
Hypotheses	sbcied.1	$$⊢\phi \to A\in V$$
	sbcied.2	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
	sbciedf.3	$$⊢Ⅎx\phi$$
	sbciedf.4	$$⊢\phi \to Ⅎx\chi$$
Assertion	sbciedf	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbcied.1	$$⊢\phi \to A\in V$$
2		sbcied.2	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3		sbciedf.3	$$⊢Ⅎx\phi$$
4		sbciedf.4	$$⊢\phi \to Ⅎx\chi$$
5	2	ex	$$⊢\phi \to \left(x=A\to \left(\psi \leftrightarrow \chi \right)\right)$$
6	3 5	alrimi	$$⊢\phi \to \forall x\left(x=A\to \left(\psi \leftrightarrow \chi \right)\right)$$
7		sbciegft	$$⊢\left(A\in V\wedge Ⅎx\chi \wedge \forall x\left(x=A\to \left(\psi \leftrightarrow \chi \right)\right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \chi \right)$$
8	1 4 6 7	syl3anc	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \chi \right)$$