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Theorem indi 3184
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 731 . . . 4  |-  ( ( x  e.  A  /\  ( x  e.  B  \/  x  e.  C
) )  <->  ( (
x  e.  A  /\  x  e.  B )  \/  ( x  e.  A  /\  x  e.  C
) ) )
2 elin 3126 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3 elin 3126 . . . . 5  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
42, 3orbi12i 681 . . . 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 176 . . 3  |-  ( ( x  e.  A  /\  ( x  e.  B  \/  x  e.  C
) )  <->  ( x  e.  ( A  i^i  B
)  \/  x  e.  ( A  i^i  C
) ) )
6 elun 3084 . . . 4  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
76anbi2i 430 . . 3  |-  ( ( x  e.  A  /\  x  e.  ( B  u.  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  \/  x  e.  C )
) )
8 elun 3084 . . 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 201 . 2  |-  ( ( x  e.  A  /\  x  e.  ( B  u.  C ) )  <->  x  e.  ( ( A  i^i  B )  u.  ( A  i^i  C ) ) )
109ineqri 3130 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 97    \/ wo 629    = wceq 1243    e. wcel 1393    u. cun 2915    i^i cin 2916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924
This theorem is referenced by:  indir  3186  undisj2  3280  disjssun  3285  difdifdirss  3307  disjpr2  3434  diftpsn3  3505  resundi  4625
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