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Theorem unundir 3758
 Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
unundir ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Proof of Theorem unundir
StepHypRef Expression
1 unidm 3739 . . 3 (𝐶𝐶) = 𝐶
21uneq2i 3747 . 2 ((𝐴𝐵) ∪ (𝐶𝐶)) = ((𝐴𝐵) ∪ 𝐶)
3 un4 3756 . 2 ((𝐴𝐵) ∪ (𝐶𝐶)) = ((𝐴𝐶) ∪ (𝐵𝐶))
42, 3eqtr3i 2650 1 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480   ∪ cun 3558 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-un 3565 This theorem is referenced by:  iocunico  37244
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