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Theorem un4 3335
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 3333 . . 3  |-  ( B  u.  ( C  u.  D ) )  =  ( C  u.  ( B  u.  D )
)
21uneq2i 3326 . 2  |-  ( A  u.  ( B  u.  ( C  u.  D
) ) )  =  ( A  u.  ( C  u.  ( B  u.  D ) ) )
3 unass 3332 . 2  |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( A  u.  ( B  u.  ( C  u.  D ) ) )
4 unass 3332 . 2  |-  ( ( A  u.  C )  u.  ( B  u.  D ) )  =  ( A  u.  ( C  u.  ( B  u.  D ) ) )
52, 3, 43eqtr4i 2313 1  |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( ( A  u.  C )  u.  ( B  u.  D )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    u. cun 3150
This theorem is referenced by:  unundi  3336  unundir  3337  xpun  4747  resasplit  5411  ex-pw  20816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157
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