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Theorem difundir 3744
Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difundir  |-  ( ( A  u.  B ) 
\  C )  =  ( ( A  \  C )  u.  ( B  \  C ) )

Proof of Theorem difundir
StepHypRef Expression
1 indir 3739 . 2  |-  ( ( A  u.  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  C ) )  u.  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3732 . 2  |-  ( ( A  u.  B )  i^i  ( _V  \  C ) )  =  ( ( A  u.  B )  \  C
)
3 invdif 3732 . . 3  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
4 invdif 3732 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
53, 4uneq12i 3649 . 2  |-  ( ( A  i^i  ( _V 
\  C ) )  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  C
)  u.  ( B 
\  C ) )
61, 2, 53eqtr3i 2497 1  |-  ( ( A  u.  B ) 
\  C )  =  ( ( A  \  C )  u.  ( B  \  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374   _Vcvv 3106    \ cdif 3466    u. cun 3467    i^i cin 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476
This theorem is referenced by:  symdif1  3756  difun2  3899  diftpsn3  4158  strleun  14574  mreexmrid  14887  mreexexlem2d  14889  mvdco  16259  dprd2da  16874  dmdprdsplit2lem  16877  ablfac1eulem  16906  lbsextlem4  17583  opsrtoslem2  17913  nulmbl2  21675  uniioombllem3  21722  ex-dif  24807  imadifxp  27117  ballotlemfp1  28056  ballotlemgun  28089  onint1  29477  fourierdlem102  31464  fourierdlem114  31476
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