Theorem iuneq12d

Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015) Remove DV conditions (Revised by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	iuneq12d.1	$$⊢\phi \to A=B$$
	iuneq12d.2	$$⊢\phi \to C=D$$
Assertion	iuneq12d	$$⊢\phi \to \bigcup _{x\in A}C=\bigcup _{x\in B}D$$

Proof

Step	Hyp	Ref	Expression
1		iuneq12d.1	$$⊢\phi \to A=B$$
2		iuneq12d.2	$$⊢\phi \to C=D$$
3	1	eleq2d	$$⊢\phi \to \left(x\in A\leftrightarrow x\in B\right)$$
4	3	anbi1d	$$⊢\phi \to \left(\left(x\in A\wedge t\in C\right)\leftrightarrow \left(x\in B\wedge t\in C\right)\right)$$
5	4	rexbidv2	$$⊢\phi \to \left(\exists x\in At\in C\leftrightarrow \exists x\in Bt\in C\right)$$
6	5	abbidv	$$⊢\phi \to \left\{t|\exists x\in At\in C\right\}=\left\{t|\exists x\in Bt\in C\right\}$$
7		df-iun	$$⊢\bigcup _{x\in A}C=\left\{t|\exists x\in At\in C\right\}$$
8		df-iun	$$⊢\bigcup _{x\in B}C=\left\{t|\exists x\in Bt\in C\right\}$$
9	6 7 8	3eqtr4g	$$⊢\phi \to \bigcup _{x\in A}C=\bigcup _{x\in B}C$$
10	2	adantr	$$⊢\left(\phi \wedge x\in B\right)\to C=D$$
11	10	iuneq2dv	$$⊢\phi \to \bigcup _{x\in B}C=\bigcup _{x\in B}D$$
12	9 11	eqtrd	$$⊢\phi \to \bigcup _{x\in A}C=\bigcup _{x\in B}D$$