Theorem iuneq1i

Description: Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021) Remove DV conditions. (Revised by GG, 1-Sep-2025)

	Ref	Expression
Hypothesis	iuneq1i.1	$$⊢A=B$$
Assertion	iuneq1i	$$⊢\bigcup _{x\in A}C=\bigcup _{x\in B}C$$

Proof

Step	Hyp	Ref	Expression
1		iuneq1i.1	$$⊢A=B$$
2	1	eleq2i	$$⊢x\in A\leftrightarrow x\in B$$
3	2	anbi1i	$$⊢\left(x\in A\wedge t\in C\right)\leftrightarrow \left(x\in B\wedge t\in C\right)$$
4	3	rexbii2	$$⊢\exists x\in At\in C\leftrightarrow \exists x\in Bt\in C$$
5	4	abbii	$$⊢\left\{t|\exists x\in At\in C\right\}=\left\{t|\exists x\in Bt\in C\right\}$$
6		df-iun	$$⊢\bigcup _{x\in A}C=\left\{t|\exists x\in At\in C\right\}$$
7		df-iun	$$⊢\bigcup _{x\in B}C=\left\{t|\exists x\in Bt\in C\right\}$$
8	5 6 7	3eqtr4i	$$⊢\bigcup _{x\in A}C=\bigcup _{x\in B}C$$