Theorem cbvcsbdavw2

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

	Ref	Expression
Hypotheses	cbvcsbdavw2.1	$$⊢\phi \to A=B$$
	cbvcsbdavw2.2	$$⊢\left(\phi \wedge x=y\right)\to C=D$$
Assertion	cbvcsbdavw2	$$⊢\phi \to ⦋A/x⦌C=⦋B/y⦌D$$

Proof

Step	Hyp	Ref	Expression
1		cbvcsbdavw2.1	$$⊢\phi \to A=B$$
2		cbvcsbdavw2.2	$$⊢\left(\phi \wedge x=y\right)\to C=D$$
3	2	eleq2d	$$⊢\left(\phi \wedge x=y\right)\to \left(t\in C\leftrightarrow t\in D\right)$$
4	1 3	cbvsbcdavw2	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}t\in C\leftrightarrow \underset{\dot{}}{[}B/y\underset{\dot{}}{]}t\in D\right)$$
5	4	abbidv	$$⊢\phi \to \left\{t|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}t\in C\right\}=\left\{t|\underset{\dot{}}{[}B/y\underset{\dot{}}{]}t\in D\right\}$$
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	3eqtr4g	$$⊢\phi \to ⦋A/x⦌C=⦋B/y⦌D$$