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Theorem difindir 3708
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 3671 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  C ) )  i^i  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3694 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  B )  \  C )
3 invdif 3694 . . 3  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
4 invdif 3694 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
53, 4ineq12i 3653 . 2  |-  ( ( A  i^i  ( _V 
\  C ) )  i^i  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  C
)  i^i  ( B  \  C ) )
61, 2, 53eqtr3i 2489 1  |-  ( ( A  i^i  B ) 
\  C )  =  ( ( A  \  C )  i^i  ( B  \  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   _Vcvv 3072    \ cdif 3428    i^i cin 3430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801  df-rab 2805  df-v 3074  df-dif 3434  df-in 3438
This theorem is referenced by:  ablfac1eulem  16690  bwthOLD  19141  ballotlemgun  27046
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