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Theorem un4 3206
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 3204 . . 3  |-  ( B  u.  ( C  u.  D ) )  =  ( C  u.  ( B  u.  D )
)
21uneq2i 3197 . 2  |-  ( A  u.  ( B  u.  ( C  u.  D
) ) )  =  ( A  u.  ( C  u.  ( B  u.  D ) ) )
3 unass 3203 . 2  |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( A  u.  ( B  u.  ( C  u.  D ) ) )
4 unass 3203 . 2  |-  ( ( A  u.  C )  u.  ( B  u.  D ) )  =  ( A  u.  ( C  u.  ( B  u.  D ) ) )
52, 3, 43eqtr4i 2148 1  |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( ( A  u.  C )  u.  ( B  u.  D )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1316    u. cun 3039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045
This theorem is referenced by:  unundi  3207  unundir  3208  xpun  4570  resasplitss  5272
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