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Theorem indir 3417
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 3415 . 2  |-  ( C  i^i  ( A  u.  B ) )  =  ( ( C  i^i  A )  u.  ( C  i^i  B ) )
2 incom 3361 . 2  |-  ( ( A  u.  B )  i^i  C )  =  ( C  i^i  ( A  u.  B )
)
3 incom 3361 . . 3  |-  ( A  i^i  C )  =  ( C  i^i  A
)
4 incom 3361 . . 3  |-  ( B  i^i  C )  =  ( C  i^i  B
)
53, 4uneq12i 3327 . 2  |-  ( ( A  i^i  C )  u.  ( B  i^i  C ) )  =  ( ( C  i^i  A
)  u.  ( C  i^i  B ) )
61, 2, 53eqtr4i 2313 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 1623    u. cun 3150    i^i cin 3151
This theorem is referenced by:  difundir  3422  undisj1  3506  resundir  4970  cdaassen  7808  fin23lem26  7951  fpwwe2lem13  8264  fiuncmp  17131  consuba  17146  trfil2  17582  tsmsres  17826  volun  18902  uniioombllem3  18940  itgsplitioo  19192  ppiprm  20389  chtprm  20391  chtdif  20396  ppidif  20401  ballotlemfp1  23050  ballotlemgun  23083  predun  24190  fixun  24449  hdrmp  25706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159
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