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Theorem difundi 3705
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 3691 . . 3  |-  ( B  u.  C )  =  ( _V  \  (
( _V  \  B
)  i^i  ( _V  \  C ) ) )
21difeq2i 3574 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( A  \  ( _V  \  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) ) )
3 inindi 3670 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( ( A  i^i  ( _V  \  B ) )  i^i  ( A  i^i  ( _V  \  C ) ) )
4 dfin2 3689 . . 3  |-  ( A  i^i  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) )  =  ( A  \  ( _V  \  ( ( _V 
\  B )  i^i  ( _V  \  C
) ) ) )
5 invdif 3694 . . . 4  |-  ( A  i^i  ( _V  \  B ) )  =  ( A  \  B
)
6 invdif 3694 . . . 4  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
75, 6ineq12i 3653 . . 3  |-  ( ( A  i^i  ( _V 
\  B ) )  i^i  ( A  i^i  ( _V  \  C ) ) )  =  ( ( A  \  B
)  i^i  ( A  \  C ) )
83, 4, 73eqtr3i 2489 . 2  |-  ( A 
\  ( _V  \ 
( ( _V  \  B )  i^i  ( _V  \  C ) ) ) )  =  ( ( A  \  B
)  i^i  ( A  \  C ) )
92, 8eqtri 2481 1  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  i^i  ( A  \  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   _Vcvv 3072    \ cdif 3428    u. cun 3429    i^i cin 3430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438
This theorem is referenced by:  undm  3711  uncld  18772  inmbl  21151  clsun  28666
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