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Theorem difundir 3422
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 3417 . 2  |-  ( ( A  u.  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  C ) )  u.  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3410 . 2  |-  ( ( A  u.  B )  i^i  ( _V  \  C ) )  =  ( ( A  u.  B )  \  C
)
3 invdif 3410 . . 3  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
4 invdif 3410 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
53, 4uneq12i 3327 . 2  |-  ( ( A  i^i  ( _V 
\  C ) )  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  C
)  u.  ( B 
\  C ) )
61, 2, 53eqtr3i 2311 1  |-  ( ( A  u.  B ) 
\  C )  =  ( ( A  \  C )  u.  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151
This theorem is referenced by:  symdif1  3433  difun2  3533  strleun  13238  mreexmrid  13545  mreexexlem2d  13547  dprd2da  15277  dmdprdsplit2lem  15280  ablfac1eulem  15307  lbsextlem4  15914  opsrtoslem2  16226  nulmbl2  18894  uniioombllem3  18940  ex-dif  20810  ballotlemfp1  23050  ballotlemgun  23083  onint1  24888  mvdco  27388  difprsneq  28068  difprsng  28069  diftpsneq  28070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159
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