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Theorem un12 3204
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 3190 . . 3 (𝐴𝐵) = (𝐵𝐴)
21uneq1i 3196 . 2 ((𝐴𝐵) ∪ 𝐶) = ((𝐵𝐴) ∪ 𝐶)
3 unass 3203 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵𝐶))
4 unass 3203 . 2 ((𝐵𝐴) ∪ 𝐶) = (𝐵 ∪ (𝐴𝐶))
52, 3, 43eqtr3i 2146 1 (𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1316  cun 3039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045
This theorem is referenced by:  un23  3205  un4  3206
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