Theorem cbviinvw2

Description: Change bound variable and domain in an indexed intersection, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbviinvw2.1	$$⊢x=y\to C=D$$
	cbviinvw2.2	$$⊢x=y\to A=B$$
Assertion	cbviinvw2	$$⊢\bigcap _{x\in A}C=\bigcap _{y\in B}D$$

Proof

Step	Hyp	Ref	Expression
1		cbviinvw2.1	$$⊢x=y\to C=D$$
2		cbviinvw2.2	$$⊢x=y\to A=B$$
3	1	eleq2d	$$⊢x=y\to \left(t\in C\leftrightarrow t\in D\right)$$
4	2 3	cbvralvw2	$$⊢\forall x\in At\in C\leftrightarrow \forall y\in Bt\in D$$
5	4	abbii	$$⊢\left\{t|\forall x\in At\in C\right\}=\left\{t|\forall y\in Bt\in D\right\}$$
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	3eqtr4i	$$⊢\bigcap _{x\in A}C=\bigcap _{y\in B}D$$