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Theorem difundi 3249
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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldif 3006 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 eldif 3006 . . . 4  |-  ( x  e.  ( A  \  C )  <->  ( x  e.  A  /\  -.  x  e.  C ) )
31, 2anbi12i 448 . . 3  |-  ( ( x  e.  ( A 
\  B )  /\  x  e.  ( A  \  C ) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  ( x  e.  A  /\  -.  x  e.  C ) ) )
4 elin 3181 . . 3  |-  ( x  e.  ( ( A 
\  B )  i^i  ( A  \  C
) )  <->  ( x  e.  ( A  \  B
)  /\  x  e.  ( A  \  C ) ) )
5 eldif 3006 . . . . . 6  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( x  e.  A  /\  -.  x  e.  ( B  u.  C
) ) )
6 elun 3139 . . . . . . . 8  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
76notbii 629 . . . . . . 7  |-  ( -.  x  e.  ( B  u.  C )  <->  -.  (
x  e.  B  \/  x  e.  C )
)
87anbi2i 445 . . . . . 6  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  u.  C )
)  <->  ( x  e.  A  /\  -.  (
x  e.  B  \/  x  e.  C )
) )
95, 8bitri 182 . . . . 5  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( x  e.  A  /\  -.  (
x  e.  B  \/  x  e.  C )
) )
10 ioran 704 . . . . . 6  |-  ( -.  ( x  e.  B  \/  x  e.  C
)  <->  ( -.  x  e.  B  /\  -.  x  e.  C ) )
1110anbi2i 445 . . . . 5  |-  ( ( x  e.  A  /\  -.  ( x  e.  B  \/  x  e.  C
) )  <->  ( x  e.  A  /\  ( -.  x  e.  B  /\  -.  x  e.  C
) ) )
129, 11bitri 182 . . . 4  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( x  e.  A  /\  ( -.  x  e.  B  /\  -.  x  e.  C
) ) )
13 anandi 557 . . . 4  |-  ( ( x  e.  A  /\  ( -.  x  e.  B  /\  -.  x  e.  C ) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  ( x  e.  A  /\  -.  x  e.  C ) ) )
1412, 13bitri 182 . . 3  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  ( x  e.  A  /\  -.  x  e.  C ) ) )
153, 4, 143bitr4ri 211 . 2  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  x  e.  ( ( A  \  B )  i^i  ( A  \  C ) ) )
1615eqriv 2085 1  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  i^i  ( A  \  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438    \ cdif 2994    u. cun 2995    i^i cin 2996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-un 3001  df-in 3003
This theorem is referenced by:  undm  3255  undifdc  6614
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