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Theorem un4 3348
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 3346 . . 3  |-  ( B  u.  ( C  u.  D ) )  =  ( C  u.  ( B  u.  D )
)
21uneq2i 3339 . 2  |-  ( A  u.  ( B  u.  ( C  u.  D
) ) )  =  ( A  u.  ( C  u.  ( B  u.  D ) ) )
3 unass 3345 . 2  |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( A  u.  ( B  u.  ( C  u.  D ) ) )
4 unass 3345 . 2  |-  ( ( A  u.  C )  u.  ( B  u.  D ) )  =  ( A  u.  ( C  u.  ( B  u.  D ) ) )
52, 3, 43eqtr4i 2326 1  |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( ( A  u.  C )  u.  ( B  u.  D )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    u. cun 3163
This theorem is referenced by:  unundi  3349  unundir  3350  xpun  4763  resasplit  5427  ex-pw  20832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170
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