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Theorem un4 3131
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 3129 . . 3  |-  ( B  u.  ( C  u.  D ) )  =  ( C  u.  ( B  u.  D )
)
21uneq2i 3122 . 2  |-  ( A  u.  ( B  u.  ( C  u.  D
) ) )  =  ( A  u.  ( C  u.  ( B  u.  D ) ) )
3 unass 3128 . 2  |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( A  u.  ( B  u.  ( C  u.  D ) ) )
4 unass 3128 . 2  |-  ( ( A  u.  C )  u.  ( B  u.  D ) )  =  ( A  u.  ( C  u.  ( B  u.  D ) ) )
52, 3, 43eqtr4i 2086 1  |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( ( A  u.  C )  u.  ( B  u.  D )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1259    u. cun 2943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950
This theorem is referenced by:  unundi  3132  unundir  3133  xpun  4429  resasplitss  5097
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