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Theorem difindi 3330
Description: Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difindi  |-  ( A 
\  ( B  i^i  C ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )

Proof of Theorem difindi
StepHypRef Expression
1 dfin3 3315 . . 3  |-  ( B  i^i  C )  =  ( _V  \  (
( _V  \  B
)  u.  ( _V 
\  C ) ) )
21difeq2i 3208 . 2  |-  ( A 
\  ( B  i^i  C ) )  =  ( A  \  ( _V 
\  ( ( _V 
\  B )  u.  ( _V  \  C
) ) ) )
3 indi 3322 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  u.  ( _V  \  C
) ) )  =  ( ( A  i^i  ( _V  \  B ) )  u.  ( A  i^i  ( _V  \  C ) ) )
4 dfin2 3312 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  u.  ( _V  \  C
) ) )  =  ( A  \  ( _V  \  ( ( _V 
\  B )  u.  ( _V  \  C
) ) ) )
5 invdif 3317 . . . 4  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
6 invdif 3317 . . . 4  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
75, 6uneq12i 3237 . . 3  |-  ( ( A  i^i  ( _V 
\  B ) )  u.  ( A  i^i  ( _V  \  C ) ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
83, 4, 73eqtr3i 2281 . 2  |-  ( A 
\  ( _V  \ 
( ( _V  \  B )  u.  ( _V  \  C ) ) ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
92, 8eqtri 2273 1  |-  ( A 
\  ( B  i^i  C ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619   _Vcvv 2727    \ cdif 3075    u. cun 3076    i^i cin 3077
This theorem is referenced by:  indm  3334  dprddisj2  15109  fctop  16573  cctop  16575  mretopd  16661  restcld  16735  cfinfil  17420  csdfil  17421  fndifnfp  25922
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085
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