Theorem sbceqbii

Description: Formula-building inference for class substitution. General version of sbcbii . (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	sbceqbii.1	$$⊢A=B$$
	sbceqbii.2	$$⊢\phi \leftrightarrow \psi$$
Assertion	sbceqbii	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\psi$$

Proof

Step	Hyp	Ref	Expression
1		sbceqbii.1	$$⊢A=B$$
2		sbceqbii.2	$$⊢\phi \leftrightarrow \psi$$
3	2	abbii	$$⊢\left\{x|\phi \right\}=\left\{x|\psi \right\}$$
4	1 3	eleq12i	$$⊢A\in \left\{x|\phi \right\}\leftrightarrow B\in \left\{x|\psi \right\}$$
5		df-sbc	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow A\in \left\{x|\phi \right\}$$
6		df-sbc	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\psi \leftrightarrow B\in \left\{x|\psi \right\}$$
7	4 5 6	3bitr4i	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\psi$$