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Theorem un12 3922
Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
un12 (𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))

Proof of Theorem un12
StepHypRef Expression
1 uncom 3908 . . 3 (𝐴𝐵) = (𝐵𝐴)
21uneq1i 3914 . 2 ((𝐴𝐵) ∪ 𝐶) = ((𝐵𝐴) ∪ 𝐶)
3 unass 3921 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵𝐶))
4 unass 3921 . 2 ((𝐵𝐴) ∪ 𝐶) = (𝐵 ∪ (𝐴𝐶))
52, 3, 43eqtr3i 2801 1 (𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cun 3721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-un 3728
This theorem is referenced by:  un23  3923  un4  3924  fresaun  6215  reconnlem1  22849  poimirlem6  33748  poimirlem7  33749  asindmre  33827  frege133d  38583
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