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Theorem indir 3581
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 3579 . 2  |-  ( C  i^i  ( A  u.  B ) )  =  ( ( C  i^i  A )  u.  ( C  i^i  B ) )
2 incom 3525 . 2  |-  ( ( A  u.  B )  i^i  C )  =  ( C  i^i  ( A  u.  B )
)
3 incom 3525 . . 3  |-  ( A  i^i  C )  =  ( C  i^i  A
)
4 incom 3525 . . 3  |-  ( B  i^i  C )  =  ( C  i^i  B
)
53, 4uneq12i 3491 . 2  |-  ( ( A  i^i  C )  u.  ( B  i^i  C ) )  =  ( ( C  i^i  A
)  u.  ( C  i^i  B ) )
61, 2, 53eqtr4i 2465 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 1652    u. cun 3310    i^i cin 3311
This theorem is referenced by:  difundir  3586  undisj1  3671  disjpr2  3862  resundir  5153  cdaassen  8052  fin23lem26  8195  fpwwe2lem13  8507  neitr  17234  fiuncmp  17457  consuba  17473  trfil2  17909  tsmsres  18163  trust  18249  restmetu  18607  volun  19429  uniioombllem3  19467  itgsplitioo  19719  ppiprm  20924  chtprm  20926  chtdif  20931  ppidif  20936  ballotlemfp1  24739  ballotlemgun  24772  predun  25450  fixun  25719  mbfposadd  26217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-in 3319
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