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Theorem unundi 3101
Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
unundi  |-  ( A  u.  ( B  u.  C ) )  =  ( ( A  u.  B )  u.  ( A  u.  C )
)

Proof of Theorem unundi
StepHypRef Expression
1 unidm 3083 . . 3  |-  ( A  u.  A )  =  A
21uneq1i 3090 . 2  |-  ( ( A  u.  A )  u.  ( B  u.  C ) )  =  ( A  u.  ( B  u.  C )
)
3 un4 3100 . 2  |-  ( ( A  u.  A )  u.  ( B  u.  C ) )  =  ( ( A  u.  B )  u.  ( A  u.  C )
)
42, 3eqtr3i 2062 1  |-  ( A  u.  ( B  u.  C ) )  =  ( ( A  u.  B )  u.  ( A  u.  C )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1243    u. cun 2912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-un 2919
This theorem is referenced by: (None)
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