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Theorem un12 3998
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 3984 . . 3 (𝐴𝐵) = (𝐵𝐴)
21uneq1i 3990 . 2 ((𝐴𝐵) ∪ 𝐶) = ((𝐵𝐴) ∪ 𝐶)
3 unass 3997 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵𝐶))
4 unass 3997 . 2 ((𝐵𝐴) ∪ 𝐶) = (𝐵 ∪ (𝐴𝐶))
52, 3, 43eqtr3i 2857 1 (𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1656  cun 3796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-un 3803
This theorem is referenced by:  un23  3999  un4  4000  fresaun  6312  reconnlem1  22999  poimirlem6  33952  poimirlem7  33953  asindmre  34031  frege133d  38891
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