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Theorem un23 3143
Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
un23 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ 𝐵)

Proof of Theorem un23
StepHypRef Expression
1 unass 3141 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵𝐶))
2 un12 3142 . 2 (𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))
3 uncom 3128 . 2 (𝐵 ∪ (𝐴𝐶)) = ((𝐴𝐶) ∪ 𝐵)
41, 2, 33eqtri 2107 1 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1285  cun 2982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988
This theorem is referenced by: (None)
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