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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 
|- ( 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 )