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Theorem difindir 3598
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 3561 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  C ) )  i^i  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3584 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  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, 4ineq12i 3542 . 2  |-  ( ( A  i^i  ( _V 
\  C ) )  i^i  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  C
)  i^i  ( B  \  C ) )
61, 2, 53eqtr3i 2466 1  |-  ( ( A  i^i  B ) 
\  C )  =  ( ( A  \  C )  i^i  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653   _Vcvv 2958    \ cdif 3319    i^i cin 3321
This theorem is referenced by:  ablfac1eulem  15635  bwth  17478  ballotlemgun  24787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-in 3329
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