Theorem sbceqbid

Description: Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018)

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

Proof

Step	Hyp	Ref	Expression
1		sbceqbid.1	$$⊢\phi \to A=B$$
2		sbceqbid.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	2	abbidv	$$⊢\phi \to \left\{x|\psi \right\}=\left\{x|\chi \right\}$$
4	1 3	eleq12d	$$⊢\phi \to \left(A\in \left\{x|\psi \right\}\leftrightarrow B\in \left\{x|\chi \right\}\right)$$
5		df-sbc	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow A\in \left\{x|\psi \right\}$$
6		df-sbc	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\chi \leftrightarrow B\in \left\{x|\chi \right\}$$
7	4 5 6	3bitr4g	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\chi \right)$$