Theorem cbviunvw2

Description: Change bound variable and domain in indexed unions, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbviunvw2.1	$$⊢x=y\to C=D$$
	cbviunvw2.2	$$⊢x=y\to A=B$$
Assertion	cbviunvw2	$$⊢\bigcup _{x\in A}C=\bigcup _{y\in B}D$$

Proof

Step	Hyp	Ref	Expression
1		cbviunvw2.1	$$⊢x=y\to C=D$$
2		cbviunvw2.2	$$⊢x=y\to A=B$$
3	1	eleq2d	$$⊢x=y\to \left(t\in C\leftrightarrow t\in D\right)$$
4	2 3	cbvrexvw2	$$⊢\exists x\in At\in C\leftrightarrow \exists y\in Bt\in D$$
5	4	abbii	$$⊢\left\{t|\exists x\in At\in C\right\}=\left\{t|\exists y\in Bt\in D\right\}$$
6		df-iun	$$⊢\bigcup _{x\in A}C=\left\{t|\exists x\in At\in C\right\}$$
7		df-iun	$$⊢\bigcup _{y\in B}D=\left\{t|\exists y\in Bt\in D\right\}$$
8	5 6 7	3eqtr4i	$$⊢\bigcup _{x\in A}C=\bigcup _{y\in B}D$$