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Theorem unundir 3280
Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
unundir  |-  ( ( A  u.  B )  u.  C )  =  ( ( A  u.  C )  u.  ( B  u.  C )
)

Proof of Theorem unundir
StepHypRef Expression
1 unidm 3261 . . 3  |-  ( C  u.  C )  =  C
21uneq2i 3269 . 2  |-  ( ( A  u.  B )  u.  ( C  u.  C ) )  =  ( ( A  u.  B )  u.  C
)
3 un4 3278 . 2  |-  ( ( A  u.  B )  u.  ( C  u.  C ) )  =  ( ( A  u.  C )  u.  ( B  u.  C )
)
42, 3eqtr3i 2187 1  |-  ( ( A  u.  B )  u.  C )  =  ( ( A  u.  C )  u.  ( B  u.  C )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1342    u. cun 3110
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-un 3116
This theorem is referenced by: (None)
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