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Theorem inindir 3551
Description: Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
inindir  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i  ( B  i^i  C ) )

Proof of Theorem inindir
StepHypRef Expression
1 inidm 3542 . . 3  |-  ( C  i^i  C )  =  C
21ineq2i 3531 . 2  |-  ( ( A  i^i  B )  i^i  ( C  i^i  C ) )  =  ( ( A  i^i  B
)  i^i  C )
3 in4 3549 . 2  |-  ( ( A  i^i  B )  i^i  ( C  i^i  C ) )  =  ( ( A  i^i  C
)  i^i  ( B  i^i  C ) )
42, 3eqtr3i 2457 1  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    i^i cin 3311
This theorem is referenced by:  difindir  3588  resindir  5155  restbas  17212  consuba  17473  kgentopon  17560  trfbas2  17865  trfil2  17909  fclsrest  18046  trust  18249  chtdif  20931  ppidif  20936  mdslmd1lem1  23818  mdslmd1lem2  23819  mddmdin0i  23924  ballotlemgun  24772  cvmsss2  24951  predin  25449
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-in 3319
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