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Theorem unundi 3288
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 3270 . . 3  |-  ( A  u.  A )  =  A
21uneq1i 3277 . 2  |-  ( ( A  u.  A )  u.  ( B  u.  C ) )  =  ( A  u.  ( B  u.  C )
)
3 un4 3287 . 2  |-  ( ( A  u.  A )  u.  ( B  u.  C ) )  =  ( ( A  u.  B )  u.  ( A  u.  C )
)
42, 3eqtr3i 2193 1  |-  ( A  u.  ( B  u.  C ) )  =  ( ( A  u.  B )  u.  ( A  u.  C )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1348    u. cun 3119
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125
This theorem is referenced by:  unfiin  6901
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