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Theorem un12 4072
 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 4058 . . 3 (𝐴𝐵) = (𝐵𝐴)
21uneq1i 4064 . 2 ((𝐴𝐵) ∪ 𝐶) = ((𝐵𝐴) ∪ 𝐶)
3 unass 4071 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵𝐶))
4 unass 4071 . 2 ((𝐵𝐴) ∪ 𝐶) = (𝐵 ∪ (𝐴𝐶))
52, 3, 43eqtr3i 2789 1 (𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∪ cun 3856 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3863 This theorem is referenced by:  un23  4073  un4  4074  fresaun  6534  unfi  8741  reconnlem1  23527  poimirlem6  35343  poimirlem7  35344  asindmre  35420  frege133d  40839
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