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Theorem un4 3499
Description: A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
un4  |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( ( A  u.  C )  u.  ( B  u.  D )
)

Proof of Theorem un4
StepHypRef Expression
1 un12 3497 . . 3  |-  ( B  u.  ( C  u.  D ) )  =  ( C  u.  ( B  u.  D )
)
21uneq2i 3490 . 2  |-  ( A  u.  ( B  u.  ( C  u.  D
) ) )  =  ( A  u.  ( C  u.  ( B  u.  D ) ) )
3 unass 3496 . 2  |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( A  u.  ( B  u.  ( C  u.  D ) ) )
4 unass 3496 . 2  |-  ( ( A  u.  C )  u.  ( B  u.  D ) )  =  ( A  u.  ( C  u.  ( B  u.  D ) ) )
52, 3, 43eqtr4i 2465 1  |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( ( A  u.  C )  u.  ( B  u.  D )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    u. cun 3310
This theorem is referenced by:  unundi  3500  unundir  3501  xpun  4927  resasplit  5605  ex-pw  21729
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317
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