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Theorem difundir 3235
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 3231 . 2  |-  ( ( A  u.  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  C ) )  u.  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3224 . 2  |-  ( ( A  u.  B )  i^i  ( _V  \  C ) )  =  ( ( A  u.  B )  \  C
)
3 invdif 3224 . . 3  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
4 invdif 3224 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
53, 4uneq12i 3136 . 2  |-  ( ( A  i^i  ( _V 
\  C ) )  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  C
)  u.  ( B 
\  C ) )
61, 2, 53eqtr3i 2111 1  |-  ( ( A  u.  B ) 
\  C )  =  ( ( A  \  C )  u.  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1285   _Vcvv 2612    \ cdif 2981    u. cun 2982    i^i cin 2983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-un 2988  df-in 2990
This theorem is referenced by:  symdif1  3247  difun2  3343  diftpsn3  3552  unfiin  6563
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