Theorem cbvsbcdavw2

Description: Change bound variable of a class substitution. General version of cbvsbcdavw . Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvsbcdavw2.1	$$⊢\phi \to A=B$$
	cbvsbcdavw2.2	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	cbvsbcdavw2	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \underset{\dot{}}{[}B/y\underset{\dot{}}{]}\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvsbcdavw2.1	$$⊢\phi \to A=B$$
2		cbvsbcdavw2.2	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
3	2	cbvabdavw	$$⊢\phi \to \left\{x|\psi \right\}=\left\{y|\chi \right\}$$
4	1 3	eleq12d	$$⊢\phi \to \left(A\in \left\{x|\psi \right\}\leftrightarrow B\in \left\{y|\chi \right\}\right)$$
5		df-sbc	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow A\in \left\{x|\psi \right\}$$
6		df-sbc	$$⊢\underset{\dot{}}{[}B/y\underset{\dot{}}{]}\chi \leftrightarrow B\in \left\{y|\chi \right\}$$
7	4 5 6	3bitr4g	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \leftrightarrow \underset{\dot{}}{[}B/y\underset{\dot{}}{]}\chi \right)$$