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 Gino Giotto, 20-Jan-2024)

	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		nfcv	$$⊢\underset{\_}{Ⅎ}yB$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}xC$$
4	2 3 1	cbviun	$$⊢\bigcup _{x\in A}B=\bigcup _{y\in A}C$$