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Theorem difindi 3727
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 3712 . . 3  |-  ( B  i^i  C )  =  ( _V  \  (
( _V  \  B
)  u.  ( _V 
\  C ) ) )
21difeq2i 3580 . 2  |-  ( A 
\  ( B  i^i  C ) )  =  ( A  \  ( _V 
\  ( ( _V 
\  B )  u.  ( _V  \  C
) ) ) )
3 indi 3719 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  u.  ( _V  \  C
) ) )  =  ( ( A  i^i  ( _V  \  B ) )  u.  ( A  i^i  ( _V  \  C ) ) )
4 dfin2 3709 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  u.  ( _V  \  C
) ) )  =  ( A  \  ( _V  \  ( ( _V 
\  B )  u.  ( _V  \  C
) ) ) )
5 invdif 3714 . . . 4  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
6 invdif 3714 . . . 4  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
75, 6uneq12i 3618 . . 3  |-  ( ( A  i^i  ( _V 
\  B ) )  u.  ( A  i^i  ( _V  \  C ) ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
83, 4, 73eqtr3i 2459 . 2  |-  ( A 
\  ( _V  \ 
( ( _V  \  B )  u.  ( _V  \  C ) ) ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
92, 8eqtri 2451 1  |-  ( A 
\  ( B  i^i  C ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   _Vcvv 3080    \ cdif 3433    u. cun 3434    i^i cin 3435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443
This theorem is referenced by:  difdif2  3730  indm  3732  fndifnfp  6108  dprddisj2  17671  fctop  20017  cctop  20019  mretopd  20106  restcld  20186  cfinfil  20906  csdfil  20907  indifundif  28151  difres  28213  unelcarsg  29152  salincl  38105
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