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Theorem un12 3754
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 3740 . . 3 (𝐴𝐵) = (𝐵𝐴)
21uneq1i 3746 . 2 ((𝐴𝐵) ∪ 𝐶) = ((𝐵𝐴) ∪ 𝐶)
3 unass 3753 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵𝐶))
4 unass 3753 . 2 ((𝐵𝐴) ∪ 𝐶) = (𝐵 ∪ (𝐴𝐶))
52, 3, 43eqtr3i 2651 1 (𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cun 3557
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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
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 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-un 3564
This theorem is referenced by:  un23  3755  un4  3756  fresaun  6037  reconnlem1  22552  poimirlem6  33082  poimirlem7  33083  asindmre  33162  frege133d  37573
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