Theorem cbviindavw2

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

	Ref	Expression
Hypotheses	cbviindavw2.1	$$⊢\left(\phi \wedge x=y\right)\to C=D$$
	cbviindavw2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
Assertion	cbviindavw2	$$⊢\phi \to \bigcap _{x\in A}C=\bigcap _{y\in B}D$$

Proof

Step	Hyp	Ref	Expression
1		cbviindavw2.1	$$⊢\left(\phi \wedge x=y\right)\to C=D$$
2		cbviindavw2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
3	1	eleq2d	$$⊢\left(\phi \wedge x=y\right)\to \left(t\in C\leftrightarrow t\in D\right)$$
4	3 2	cbvraldva2	$$⊢\phi \to \left(\forall x\in At\in C\leftrightarrow \forall y\in Bt\in D\right)$$
5	4	abbidv	$$⊢\phi \to \left\{t|\forall x\in At\in C\right\}=\left\{t|\forall y\in Bt\in D\right\}$$
6		df-iin	$$⊢\bigcap _{x\in A}C=\left\{t|\forall x\in At\in C\right\}$$
7		df-iin	$$⊢\bigcap _{y\in B}D=\left\{t|\forall y\in Bt\in D\right\}$$
8	5 6 7	3eqtr4g	$$⊢\phi \to \bigcap _{x\in A}C=\bigcap _{y\in B}D$$