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, 14-Aug-2025)

	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	1	eleq2d	$$⊢x=y\to \left(z\in B\leftrightarrow z\in C\right)$$
3	2	cbvralvw	$$⊢\forall x\in Az\in B\leftrightarrow \forall y\in Az\in C$$
4	3	abbii	$$⊢\left\{z|\forall x\in Az\in B\right\}=\left\{z|\forall y\in Az\in C\right\}$$
5		df-iin	$$⊢\bigcap _{x\in A}B=\left\{z|\forall x\in Az\in B\right\}$$
6		df-iin	$$⊢\bigcap _{y\in A}C=\left\{z|\forall y\in Az\in C\right\}$$
7	4 5 6	3eqtr4i	$$⊢\bigcap _{x\in A}B=\bigcap _{y\in A}C$$