Theorem cbvsbcvw2

Description: Change bound variable of a class substitution using implicit substitution. General version of cbvsbcvw . (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	cbvsbcvw2.1	$$⊢A=B$$
	cbvsbcvw2.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvsbcvw2	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}B/y\underset{\dot{}}{]}\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvsbcvw2.1	$$⊢A=B$$
2		cbvsbcvw2.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3	2	cbvabv	$$⊢\left\{x|\phi \right\}=\left\{y|\psi \right\}$$
4	1 3	eleq12i	$$⊢A\in \left\{x|\phi \right\}\leftrightarrow B\in \left\{y|\psi \right\}$$
5		df-sbc	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow A\in \left\{x|\phi \right\}$$
6		df-sbc	$$⊢\underset{\dot{}}{[}B/y\underset{\dot{}}{]}\psi \leftrightarrow B\in \left\{y|\psi \right\}$$
7	4 5 6	3bitr4i	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}B/y\underset{\dot{}}{]}\psi$$