Theorem cbviindavw

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

	Ref	Expression
Hypothesis	cbviindavw.1	$$⊢\left(\phi \wedge x=y\right)\to B=C$$
Assertion	cbviindavw	$$⊢\phi \to \bigcap _{x\in A}B=\bigcap _{y\in A}C$$

Proof

Step	Hyp	Ref	Expression
1		cbviindavw.1	$$⊢\left(\phi \wedge x=y\right)\to B=C$$
2	1	eleq2d	$$⊢\left(\phi \wedge x=y\right)\to \left(t\in B\leftrightarrow t\in C\right)$$
3	2	cbvraldva	$$⊢\phi \to \left(\forall x\in At\in B\leftrightarrow \forall y\in At\in C\right)$$
4	3	abbidv	$$⊢\phi \to \left\{t|\forall x\in At\in B\right\}=\left\{t|\forall y\in At\in C\right\}$$
5		df-iin	$$⊢\bigcap _{x\in A}B=\left\{t|\forall x\in At\in B\right\}$$
6		df-iin	$$⊢\bigcap _{y\in A}C=\left\{t|\forall y\in At\in C\right\}$$
7	4 5 6	3eqtr4g	$$⊢\phi \to \bigcap _{x\in A}B=\bigcap _{y\in A}C$$