Theorem cbvcsbdavw

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

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

Proof

Step	Hyp	Ref	Expression
1		cbvcsbdavw.1	$$⊢\left(\phi \wedge x=y\right)\to B=C$$
2	1	eleq2d	$$⊢\left(\phi \wedge x=y\right)\to \left(t\in B\leftrightarrow t\in C\right)$$
3	2	cbvsbcdavw	$$⊢\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}t\in B\leftrightarrow \underset{\dot{}}{[}A/y\underset{\dot{}}{]}t\in C\right)$$
4	3	abbidv	$$⊢\phi \to \left\{t|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}t\in B\right\}=\left\{t|\underset{\dot{}}{[}A/y\underset{\dot{}}{]}t\in C\right\}$$
5		df-csb	$$⊢⦋A/x⦌B=\left\{t|\underset{\dot{}}{[}A/x\underset{\dot{}}{]}t\in B\right\}$$
6		df-csb	$$⊢⦋A/y⦌C=\left\{t|\underset{\dot{}}{[}A/y\underset{\dot{}}{]}t\in C\right\}$$
7	4 5 6	3eqtr4g	$$⊢\phi \to ⦋A/x⦌B=⦋A/y⦌C$$