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Theorem unundir 3918
 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 3899 . . 3 (𝐶𝐶) = 𝐶
21uneq2i 3907 . 2 ((𝐴𝐵) ∪ (𝐶𝐶)) = ((𝐴𝐵) ∪ 𝐶)
3 un4 3916 . 2 ((𝐴𝐵) ∪ (𝐶𝐶)) = ((𝐴𝐶) ∪ (𝐵𝐶))
42, 3eqtr3i 2784 1 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∪ cun 3713 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720 This theorem is referenced by:  iocunico  38298
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