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Theorem difundi 3396
Description: Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difundi  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  i^i  ( A  \  C ) )

Proof of Theorem difundi
StepHypRef Expression
1 dfun3 3382 . . 3  |-  ( B  u.  C )  =  ( _V  \  (
( _V  \  B
)  i^i  ( _V  \  C ) ) )
21difeq2i 3266 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( A  \  ( _V  \  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) ) )
3 inindi 3361 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( ( A  i^i  ( _V  \  B ) )  i^i  ( A  i^i  ( _V  \  C ) ) )
4 dfin2 3380 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( A  \  ( _V  \  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) ) )
5 invdif 3385 . . . 4  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
6 invdif 3385 . . . 4  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
75, 6ineq12i 3343 . . 3  |-  ( ( A  i^i  ( _V 
\  B ) )  i^i  ( A  i^i  ( _V  \  C ) ) )  =  ( ( A  \  B
)  i^i  ( A  \  C ) )
83, 4, 73eqtr3i 2286 . 2  |-  ( A 
\  ( _V  \ 
( ( _V  \  B )  i^i  ( _V  \  C ) ) ) )  =  ( ( A  \  B
)  i^i  ( A  \  C ) )
92, 8eqtri 2278 1  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  i^i  ( A  \  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619   _Vcvv 2763    \ cdif 3124    u. cun 3125    i^i cin 3126
This theorem is referenced by:  undm  3401  uncld  16740  inmbl  18861  clsun  25613
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 1927  ax-ext 2239
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 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134
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