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Theorem indir 3359
Description: Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
indir  |-  ( ( A  u.  B )  i^i  C )  =  ( ( A  i^i  C )  u.  ( B  i^i  C ) )

Proof of Theorem indir
StepHypRef Expression
1 indi 3357 . 2  |-  ( C  i^i  ( A  u.  B ) )  =  ( ( C  i^i  A )  u.  ( C  i^i  B ) )
2 incom 3303 . 2  |-  ( ( A  u.  B )  i^i  C )  =  ( C  i^i  ( A  u.  B )
)
3 incom 3303 . . 3  |-  ( A  i^i  C )  =  ( C  i^i  A
)
4 incom 3303 . . 3  |-  ( B  i^i  C )  =  ( C  i^i  B
)
53, 4uneq12i 3269 . 2  |-  ( ( A  i^i  C )  u.  ( B  i^i  C ) )  =  ( ( C  i^i  A
)  u.  ( C  i^i  B ) )
61, 2, 53eqtr4i 2286 1  |-  ( ( A  u.  B )  i^i  C )  =  ( ( A  i^i  C )  u.  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    u. cun 3092    i^i cin 3093
This theorem is referenced by:  difundir  3364  undisj1  3448  resundir  4923  cdaassen  7741  fin23lem26  7884  fpwwe2lem13  8197  fiuncmp  17058  consuba  17073  trfil2  17509  tsmsres  17753  volun  18829  uniioombllem3  18867  itgsplitioo  19119  ppiprm  20316  chtprm  20318  chtdif  20323  ppidif  20328  ballotlemfp1  22976  ballotlemgun  23009  predun  23524  fixun  23790  hdrmp  25038
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-un 3099  df-in 3101
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