Theorem cbviunv

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003) Add disjoint variable condition to avoid ax-13 . See cbviunvg for a less restrictive version requiring more axioms. (Revised by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbviunv.1	$$⊢x=y\to B=C$$
Assertion	cbviunv	$$⊢\bigcup _{x\in A}B=\bigcup _{y\in A}C$$

Proof

Step	Hyp	Ref	Expression
1		cbviunv.1	$$⊢x=y\to B=C$$
2	1	eleq2d	$$⊢x=y\to \left(z\in B\leftrightarrow z\in C\right)$$
3	2	cbvrexvw	$$⊢\exists x\in Az\in B\leftrightarrow \exists y\in Az\in C$$
4	3	abbii	$$⊢\left\{z|\exists x\in Az\in B\right\}=\left\{z|\exists y\in Az\in C\right\}$$
5		df-iun	$$⊢\bigcup _{x\in A}B=\left\{z|\exists x\in Az\in B\right\}$$
6		df-iun	$$⊢\bigcup _{y\in A}C=\left\{z|\exists y\in Az\in C\right\}$$
7	4 5 6	3eqtr4i	$$⊢\bigcup _{x\in A}B=\bigcup _{y\in A}C$$