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Theorem difundir 3596
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 3591 . 2  |-  ( ( A  u.  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  C ) )  u.  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3584 . 2  |-  ( ( A  u.  B )  i^i  ( _V  \  C ) )  =  ( ( A  u.  B )  \  C
)
3 invdif 3584 . . 3  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
4 invdif 3584 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
53, 4uneq12i 3501 . 2  |-  ( ( A  i^i  ( _V 
\  C ) )  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  C
)  u.  ( B 
\  C ) )
61, 2, 53eqtr3i 2466 1  |-  ( ( A  u.  B ) 
\  C )  =  ( ( A  \  C )  u.  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653   _Vcvv 2958    \ cdif 3319    u. cun 3320    i^i cin 3321
This theorem is referenced by:  symdif1  3608  difun2  3709  diftpsn3  3939  strleun  13561  mreexmrid  13870  mreexexlem2d  13872  dprd2da  15602  dmdprdsplit2lem  15605  ablfac1eulem  15632  lbsextlem4  16235  opsrtoslem2  16547  nulmbl2  19433  uniioombllem3  19479  ex-dif  21733  imadifxp  24040  ballotlemfp1  24751  ballotlemgun  24784  onint1  26201  mvdco  27367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329
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