Theorem cbvcsbv

Description: Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008) (Revised by Mario Carneiro, 13-Oct-2016)

	Ref	Expression
Hypothesis	cbvcsbv.1	$$⊢x=y\to B=C$$
Assertion	cbvcsbv	$$⊢⦋A/x⦌B=⦋A/y⦌C$$

Proof

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