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Theorem difindir 3500
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 3463 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  C ) )  i^i  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3486 . 2  |-  ( ( A  i^i  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  B )  \  C )
3 invdif 3486 . . 3  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
4 invdif 3486 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
53, 4ineq12i 3444 . 2  |-  ( ( A  i^i  ( _V 
\  C ) )  i^i  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  C
)  i^i  ( B  \  C ) )
61, 2, 53eqtr3i 2386 1  |-  ( ( A  i^i  B ) 
\  C )  =  ( ( A  \  C )  i^i  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1642   _Vcvv 2864    \ cdif 3225    i^i cin 3227
This theorem is referenced by:  ablfac1eulem  15406  ballotlemgun  24031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rab 2628  df-v 2866  df-dif 3231  df-in 3235
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