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Theorem difindir 3540
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 3503 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  C ) )  i^i  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3526 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  B )  \  C )
3 invdif 3526 . . 3  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
4 invdif 3526 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
53, 4ineq12i 3484 . 2  |-  ( ( A  i^i  ( _V 
\  C ) )  i^i  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  C
)  i^i  ( B  \  C ) )
61, 2, 53eqtr3i 2416 1  |-  ( ( A  i^i  B ) 
\  C )  =  ( ( A  \  C )  i^i  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649   _Vcvv 2900    \ cdif 3261    i^i cin 3263
This theorem is referenced by:  ablfac1eulem  15558  ballotlemgun  24562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rab 2659  df-v 2902  df-dif 3267  df-in 3271
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