Theorem cbviundavw2

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

	Ref	Expression
Hypotheses	cbviundavw2.1	$$⊢\left(\phi \wedge x=y\right)\to C=D$$
	cbviundavw2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
Assertion	cbviundavw2	$$⊢\phi \to \bigcup _{x\in A}C=\bigcup _{y\in B}D$$

Proof

Step	Hyp	Ref	Expression
1		cbviundavw2.1	$$⊢\left(\phi \wedge x=y\right)\to C=D$$
2		cbviundavw2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
3	1	eleq2d	$$⊢\left(\phi \wedge x=y\right)\to \left(t\in C\leftrightarrow t\in D\right)$$
4	3 2	cbvrexdva2	$$⊢\phi \to \left(\exists x\in At\in C\leftrightarrow \exists y\in Bt\in D\right)$$
5	4	abbidv	$$⊢\phi \to \left\{t|\exists x\in At\in C\right\}=\left\{t|\exists y\in Bt\in D\right\}$$
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	3eqtr4g	$$⊢\phi \to \bigcup _{x\in A}C=\bigcup _{y\in B}D$$