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Theorem difundi 3333
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 3085 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 eldif 3085 . . . 4  |-  ( x  e.  ( A  \  C )  <->  ( x  e.  A  /\  -.  x  e.  C ) )
31, 2anbi12i 456 . . 3  |-  ( ( x  e.  ( A 
\  B )  /\  x  e.  ( A  \  C ) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  ( x  e.  A  /\  -.  x  e.  C ) ) )
4 elin 3264 . . 3  |-  ( x  e.  ( ( A 
\  B )  i^i  ( A  \  C
) )  <->  ( x  e.  ( A  \  B
)  /\  x  e.  ( A  \  C ) ) )
5 eldif 3085 . . . . . 6  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( x  e.  A  /\  -.  x  e.  ( B  u.  C
) ) )
6 elun 3222 . . . . . . . 8  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
76notbii 658 . . . . . . 7  |-  ( -.  x  e.  ( B  u.  C )  <->  -.  (
x  e.  B  \/  x  e.  C )
)
87anbi2i 453 . . . . . 6  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  u.  C )
)  <->  ( x  e.  A  /\  -.  (
x  e.  B  \/  x  e.  C )
) )
95, 8bitri 183 . . . . 5  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( x  e.  A  /\  -.  (
x  e.  B  \/  x  e.  C )
) )
10 ioran 742 . . . . . 6  |-  ( -.  ( x  e.  B  \/  x  e.  C
)  <->  ( -.  x  e.  B  /\  -.  x  e.  C ) )
1110anbi2i 453 . . . . 5  |-  ( ( x  e.  A  /\  -.  ( x  e.  B  \/  x  e.  C
) )  <->  ( x  e.  A  /\  ( -.  x  e.  B  /\  -.  x  e.  C
) ) )
129, 11bitri 183 . . . 4  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( x  e.  A  /\  ( -.  x  e.  B  /\  -.  x  e.  C
) ) )
13 anandi 580 . . . 4  |-  ( ( x  e.  A  /\  ( -.  x  e.  B  /\  -.  x  e.  C ) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  ( x  e.  A  /\  -.  x  e.  C ) ) )
1412, 13bitri 183 . . 3  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  ( x  e.  A  /\  -.  x  e.  C ) ) )
153, 4, 143bitr4ri 212 . 2  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  x  e.  ( ( A  \  B )  i^i  ( A  \  C ) ) )
1615eqriv 2137 1  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  i^i  ( A  \  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    \/ wo 698    = wceq 1332    e. wcel 1481    \ cdif 3073    u. cun 3074    i^i cin 3075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-un 3080  df-in 3082
This theorem is referenced by:  undm  3339  undifdc  6820  uncld  12321
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