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Theorem un23 3506
Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
un23  |-  ( ( A  u.  B )  u.  C )  =  ( ( A  u.  C )  u.  B
)

Proof of Theorem un23
StepHypRef Expression
1 unass 3504 . 2  |-  ( ( A  u.  B )  u.  C )  =  ( A  u.  ( B  u.  C )
)
2 un12 3505 . 2  |-  ( A  u.  ( B  u.  C ) )  =  ( B  u.  ( A  u.  C )
)
3 uncom 3491 . 2  |-  ( B  u.  ( A  u.  C ) )  =  ( ( A  u.  C )  u.  B
)
41, 2, 33eqtri 2460 1  |-  ( ( A  u.  B )  u.  C )  =  ( ( A  u.  C )  u.  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    u. cun 3318
This theorem is referenced by:  ssunpr  3961  setscom  13497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325
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