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Theorem difindir 3301
Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difindir  |-  ( ( A  i^i  B ) 
\  C )  =  ( ( A  \  C )  i^i  ( B  \  C ) )

Proof of Theorem difindir
StepHypRef Expression
1 inindir 3264 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  C ) )  i^i  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3288 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  B )  \  C )
3 invdif 3288 . . 3  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
4 invdif 3288 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
53, 4ineq12i 3245 . 2  |-  ( ( A  i^i  ( _V 
\  C ) )  i^i  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  C
)  i^i  ( B  \  C ) )
61, 2, 53eqtr3i 2146 1  |-  ( ( A  i^i  B ) 
\  C )  =  ( ( A  \  C )  i^i  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1316   _Vcvv 2660    \ cdif 3038    i^i cin 3040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-in 3047
This theorem is referenced by: (None)
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