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Theorem un23 4173
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 4171 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵𝐶))
2 un12 4172 . 2 (𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))
3 uncom 4157 . 2 (𝐵 ∪ (𝐴𝐶)) = ((𝐴𝐶) ∪ 𝐵)
41, 2, 33eqtri 2768 1 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cun 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955
This theorem is referenced by:  ssunpr  4833  setscom  17218  cycpmco2rn  33146  poimirlem6  37634  poimirlem7  37635  poimirlem16  37644  poimirlem19  37647  iocunico  43228  dfrcl2  43692
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