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Theorem indi 3287
Description: Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
indi  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )

Proof of Theorem indi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 andi 790 . . . 4  |-  ( ( x  e.  A  /\  ( x  e.  B  \/  x  e.  C
) )  <->  ( (
x  e.  A  /\  x  e.  B )  \/  ( x  e.  A  /\  x  e.  C
) ) )
2 elin 3223 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3 elin 3223 . . . . 5  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
42, 3orbi12i 736 . . . 4  |-  ( ( x  e.  ( A  i^i  B )  \/  x  e.  ( A  i^i  C ) )  <-> 
( ( x  e.  A  /\  x  e.  B )  \/  (
x  e.  A  /\  x  e.  C )
) )
51, 4bitr4i 186 . . 3  |-  ( ( x  e.  A  /\  ( x  e.  B  \/  x  e.  C
) )  <->  ( x  e.  ( A  i^i  B
)  \/  x  e.  ( A  i^i  C
) ) )
6 elun 3181 . . . 4  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
76anbi2i 450 . . 3  |-  ( ( x  e.  A  /\  x  e.  ( B  u.  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  \/  x  e.  C )
) )
8 elun 3181 . . 3  |-  ( x  e.  ( ( A  i^i  B )  u.  ( A  i^i  C
) )  <->  ( x  e.  ( A  i^i  B
)  \/  x  e.  ( A  i^i  C
) ) )
95, 7, 83bitr4i 211 . 2  |-  ( ( x  e.  A  /\  x  e.  ( B  u.  C ) )  <->  x  e.  ( ( A  i^i  B )  u.  ( A  i^i  C ) ) )
109ineqri 3233 1  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    \/ wo 680    = wceq 1312    e. wcel 1461    u. cun 3033    i^i cin 3034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-in 3041
This theorem is referenced by:  indir  3289  undisj2  3385  disjssun  3390  difdifdirss  3411  disjpr2  3551  diftpsn3  3625  resundi  4788
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