Theorem cbvcsbvw2

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

	Ref	Expression
Hypotheses	cbvcsbvw2.1	$$⊢A=B$$
	cbvcsbvw2.2	$$⊢x=y\to C=D$$
Assertion	cbvcsbvw2	$$⊢⦋A/x⦌C=⦋B/y⦌D$$

Proof

Step	Hyp	Ref	Expression
1		cbvcsbvw2.1	$$⊢A=B$$
2		cbvcsbvw2.2	$$⊢x=y\to C=D$$
3	2	eleq2d	$$⊢x=y\to \left(t\in C\leftrightarrow t\in D\right)$$
4	1 3	cbvsbcvw2	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}t\in C\leftrightarrow \underset{\dot{}}{[}B/y\underset{\dot{}}{]}t\in D$$
5	4	abbii	$$⊢\left\{t|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}t\in C\right\}=\left\{t|\underset{\dot{}}{[}B/y\underset{\dot{}}{]}t\in D\right\}$$
6		df-csb	$$⊢⦋A/x⦌C=\left\{t|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}t\in C\right\}$$
7		df-csb	$$⊢⦋B/y⦌D=\left\{t|\underset{\dot{}}{[}B/y\underset{\dot{}}{]}t\in D\right\}$$
8	5 6 7	3eqtr4i	$$⊢⦋A/x⦌C=⦋B/y⦌D$$