Theorem sbc2ie

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008) (Revised by Mario Carneiro, 19-Dec-2013) (Proof shortened by Gino Giotto, 12-Oct-2024)

	Ref	Expression
Hypotheses	sbc2ie.1	$$⊢A\in \mathrm{V}$$
	sbc2ie.2	$$⊢B\in \mathrm{V}$$
	sbc2ie.3	$$⊢\left(x=A\wedge y=B\right)\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	sbc2ie	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\underset{\dot{}}{[}B/y\underset{\dot{}}{]}\phi \leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		sbc2ie.1	$$⊢A\in \mathrm{V}$$
2		sbc2ie.2	$$⊢B\in \mathrm{V}$$
3		sbc2ie.3	$$⊢\left(x=A\wedge y=B\right)\to \left(\phi \leftrightarrow \psi \right)$$
4	2	a1i	$$⊢x=A\to B\in \mathrm{V}$$
5	4 3	sbcied	$$⊢x=A\to \left(\underset{\dot{}}{[}B/y\underset{\dot{}}{]}\phi \leftrightarrow \psi \right)$$
6	1 5	sbcie	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\underset{\dot{}}{[}B/y\underset{\dot{}}{]}\phi \leftrightarrow \psi$$