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Theorem unundir 4171
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 4152 . . 3 (𝐶𝐶) = 𝐶
21uneq2i 4160 . 2 ((𝐴𝐵) ∪ (𝐶𝐶)) = ((𝐴𝐵) ∪ 𝐶)
3 un4 4169 . 2 ((𝐴𝐵) ∪ (𝐶𝐶)) = ((𝐴𝐶) ∪ (𝐵𝐶))
42, 3eqtr3i 2761 1 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cun 3946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3953
This theorem is referenced by:  iocunico  42425
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