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Theorem difundi 3415
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 3166 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 eldif 3166 . . . 4  |-  ( x  e.  ( A  \  C )  <->  ( x  e.  A  /\  -.  x  e.  C ) )
31, 2anbi12i 460 . . 3  |-  ( ( x  e.  ( A 
\  B )  /\  x  e.  ( A  \  C ) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  ( x  e.  A  /\  -.  x  e.  C ) ) )
4 elin 3346 . . 3  |-  ( x  e.  ( ( A 
\  B )  i^i  ( A  \  C
) )  <->  ( x  e.  ( A  \  B
)  /\  x  e.  ( A  \  C ) ) )
5 eldif 3166 . . . . . 6  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( x  e.  A  /\  -.  x  e.  ( B  u.  C
) ) )
6 elun 3304 . . . . . . . 8  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
76notbii 669 . . . . . . 7  |-  ( -.  x  e.  ( B  u.  C )  <->  -.  (
x  e.  B  \/  x  e.  C )
)
87anbi2i 457 . . . . . 6  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  u.  C )
)  <->  ( x  e.  A  /\  -.  (
x  e.  B  \/  x  e.  C )
) )
95, 8bitri 184 . . . . 5  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( x  e.  A  /\  -.  (
x  e.  B  \/  x  e.  C )
) )
10 ioran 753 . . . . . 6  |-  ( -.  ( x  e.  B  \/  x  e.  C
)  <->  ( -.  x  e.  B  /\  -.  x  e.  C ) )
1110anbi2i 457 . . . . 5  |-  ( ( x  e.  A  /\  -.  ( x  e.  B  \/  x  e.  C
) )  <->  ( x  e.  A  /\  ( -.  x  e.  B  /\  -.  x  e.  C
) ) )
129, 11bitri 184 . . . 4  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( x  e.  A  /\  ( -.  x  e.  B  /\  -.  x  e.  C
) ) )
13 anandi 590 . . . 4  |-  ( ( x  e.  A  /\  ( -.  x  e.  B  /\  -.  x  e.  C ) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  ( x  e.  A  /\  -.  x  e.  C ) ) )
1412, 13bitri 184 . . 3  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  ( x  e.  A  /\  -.  x  e.  C ) ) )
153, 4, 143bitr4ri 213 . 2  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  x  e.  ( ( A  \  B )  i^i  ( A  \  C ) ) )
1615eqriv 2193 1  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  i^i  ( A  \  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    \/ wo 709    = wceq 1364    e. wcel 2167    \ cdif 3154    u. cun 3155    i^i cin 3156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-in 3163
This theorem is referenced by:  undm  3421  undifdc  6985  uncld  14349
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