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Theorem difindi 3555
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 3540 . . 3  |-  ( B  i^i  C )  =  ( _V  \  (
( _V  \  B
)  u.  ( _V 
\  C ) ) )
21difeq2i 3422 . 2  |-  ( A 
\  ( B  i^i  C ) )  =  ( A  \  ( _V 
\  ( ( _V 
\  B )  u.  ( _V  \  C
) ) ) )
3 indi 3547 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  u.  ( _V  \  C
) ) )  =  ( ( A  i^i  ( _V  \  B ) )  u.  ( A  i^i  ( _V  \  C ) ) )
4 dfin2 3537 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  u.  ( _V  \  C
) ) )  =  ( A  \  ( _V  \  ( ( _V 
\  B )  u.  ( _V  \  C
) ) ) )
5 invdif 3542 . . . 4  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
6 invdif 3542 . . . 4  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
75, 6uneq12i 3459 . . 3  |-  ( ( A  i^i  ( _V 
\  B ) )  u.  ( A  i^i  ( _V  \  C ) ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
83, 4, 73eqtr3i 2432 . 2  |-  ( A 
\  ( _V  \ 
( ( _V  \  B )  u.  ( _V  \  C ) ) ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
92, 8eqtri 2424 1  |-  ( A 
\  ( B  i^i  C ) )  =  ( ( A  \  B
)  u.  ( A 
\  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279
This theorem is referenced by:  difdif2  3558  indm  3560  dprddisj2  15552  fctop  17023  cctop  17025  mretopd  17111  restcld  17190  cfinfil  17878  csdfil  17879  fndifnfp  26627
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287
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