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Metamath Proof Explorer

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	$⊢ φ \to A = B$
		iuneq12d.2	$⊢ φ \to C = D$
	Assertion	iuneq12d	$⊢ φ \to ⋃_{x \in A} C = ⋃_{x \in B} D$

Proof

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