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Theorem difundi 3217
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 2955 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 eldif 2955 . . . 4  |-  ( x  e.  ( A  \  C )  <->  ( x  e.  A  /\  -.  x  e.  C ) )
31, 2anbi12i 441 . . 3  |-  ( ( x  e.  ( A 
\  B )  /\  x  e.  ( A  \  C ) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  ( x  e.  A  /\  -.  x  e.  C ) ) )
4 elin 3154 . . 3  |-  ( x  e.  ( ( A 
\  B )  i^i  ( A  \  C
) )  <->  ( x  e.  ( A  \  B
)  /\  x  e.  ( A  \  C ) ) )
5 eldif 2955 . . . . . 6  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( x  e.  A  /\  -.  x  e.  ( B  u.  C
) ) )
6 elun 3112 . . . . . . . 8  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
76notbii 604 . . . . . . 7  |-  ( -.  x  e.  ( B  u.  C )  <->  -.  (
x  e.  B  \/  x  e.  C )
)
87anbi2i 438 . . . . . 6  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  u.  C )
)  <->  ( x  e.  A  /\  -.  (
x  e.  B  \/  x  e.  C )
) )
95, 8bitri 177 . . . . 5  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( x  e.  A  /\  -.  (
x  e.  B  \/  x  e.  C )
) )
10 ioran 679 . . . . . 6  |-  ( -.  ( x  e.  B  \/  x  e.  C
)  <->  ( -.  x  e.  B  /\  -.  x  e.  C ) )
1110anbi2i 438 . . . . 5  |-  ( ( x  e.  A  /\  -.  ( x  e.  B  \/  x  e.  C
) )  <->  ( x  e.  A  /\  ( -.  x  e.  B  /\  -.  x  e.  C
) ) )
129, 11bitri 177 . . . 4  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( x  e.  A  /\  ( -.  x  e.  B  /\  -.  x  e.  C
) ) )
13 anandi 532 . . . 4  |-  ( ( x  e.  A  /\  ( -.  x  e.  B  /\  -.  x  e.  C ) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  ( x  e.  A  /\  -.  x  e.  C ) ) )
1412, 13bitri 177 . . 3  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  ( x  e.  A  /\  -.  x  e.  C ) ) )
153, 4, 143bitr4ri 206 . 2  |-  ( x  e.  ( A  \ 
( B  u.  C
) )  <->  x  e.  ( ( A  \  B )  i^i  ( A  \  C ) ) )
1615eqriv 2053 1  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  i^i  ( A  \  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 101    \/ wo 639    = wceq 1259    e. wcel 1409    \ cdif 2942    u. cun 2943    i^i cin 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-un 2950  df-in 2952
This theorem is referenced by:  undm  3223
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