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	|- ( ph -> A = B )
	iuneq12d.2	|- ( ph -> C = D )
Assertion	iuneq12d	|- ( ph -> U_ x e. A C = U_ x e. B D )

Proof

Step	Hyp	Ref	Expression
1		iuneq12d.1	|- ( ph -> A = B )
2		iuneq12d.2	|- ( ph -> C = D )
3	1	eleq2d	|- ( ph -> ( x e. A <-> x e. B ) )
4	3	anbi1d	|- ( ph -> ( ( x e. A /\ t e. C ) <-> ( x e. B /\ t e. C ) ) )
5	4	rexbidv2	|- ( ph -> ( E. x e. A t e. C <-> E. x e. B t e. C ) )
6	5	abbidv	|- ( ph -> { t | E. x e. A t e. C } = { t | E. x e. B t e. C } )
7		df-iun	|- U_ x e. A C = { t | E. x e. A t e. C }
8		df-iun	|- U_ x e. B C = { t | E. x e. B t e. C }
9	6 7 8	3eqtr4g	|- ( ph -> U_ x e. A C = U_ x e. B C )
10	2	adantr	|- ( ( ph /\ x e. B ) -> C = D )
11	10	iuneq2dv	|- ( ph -> U_ x e. B C = U_ x e. B D )
12	9 11	eqtrd	|- ( ph -> U_ x e. A C = U_ x e. B D )