Theorem cbviinv

Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009) Add disjoint variable condition to avoid ax-13 . See cbviinvg for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024)

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

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	cbviin	$$⊢\bigcap _{x\in A}B=\bigcap _{y\in A}C$$