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Theorem unundi 3650
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 3632 . . 3  |-  ( A  u.  A )  =  A
21uneq1i 3639 . 2  |-  ( ( A  u.  A )  u.  ( B  u.  C ) )  =  ( A  u.  ( B  u.  C )
)
3 un4 3649 . 2  |-  ( ( A  u.  A )  u.  ( B  u.  C ) )  =  ( ( A  u.  B )  u.  ( A  u.  C )
)
42, 3eqtr3i 2474 1  |-  ( A  u.  ( B  u.  C ) )  =  ( ( A  u.  B )  u.  ( A  u.  C )
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383    u. cun 3459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-un 3466
This theorem is referenced by:  dfif5  3942
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