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Theorem inass 3495
Description: Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
Assertion
Ref Expression
inass  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )

Proof of Theorem inass
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 anass 631 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  x  e.  C )  <->  ( x  e.  A  /\  (
x  e.  B  /\  x  e.  C )
) )
2 elin 3474 . . . . 5  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
32anbi2i 676 . . . 4  |-  ( ( x  e.  A  /\  x  e.  ( B  i^i  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  /\  x  e.  C )
) )
41, 3bitr4i 244 . . 3  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  x  e.  C )  <->  ( x  e.  A  /\  x  e.  ( B  i^i  C
) ) )
5 elin 3474 . . . 4  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
65anbi1i 677 . . 3  |-  ( ( x  e.  ( A  i^i  B )  /\  x  e.  C )  <->  ( ( x  e.  A  /\  x  e.  B
)  /\  x  e.  C ) )
7 elin 3474 . . 3  |-  ( x  e.  ( A  i^i  ( B  i^i  C ) )  <->  ( x  e.  A  /\  x  e.  ( B  i^i  C
) ) )
84, 6, 73bitr4i 269 . 2  |-  ( ( x  e.  ( A  i^i  B )  /\  x  e.  C )  <->  x  e.  ( A  i^i  ( B  i^i  C ) ) )
98ineqri 3478 1  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717    i^i cin 3263
This theorem is referenced by:  in12  3496  in32  3497  in4  3501  indif2  3528  difun1  3545  dfrab3ss  3563  dfif4  3694  onfr  4562  resres  5100  inres  5105  imainrect  5253  fresaun  5555  fresaunres2  5556  epfrs  7601  incexclem  12544  sadeq  12912  smuval2  12922  smumul  12933  ressinbas  13453  ressress  13454  resscatc  14188  sylow2a  15181  ablfac1eu  15559  ressmplbas2  16446  restco  17151  restopnb  17162  kgeni  17491  hausdiag  17599  fclsrest  17978  clsocv  19076  itg2cnlem2  19522  rplogsum  21089  chjassi  22837  pjoml2i  22936  cmcmlem  22942  cmbr3i  22951  fh1  22969  fh2  22970  pj3lem1  23558  dmdbr5  23660  mdslmd3i  23684  mdexchi  23687  atabsi  23753  dmdbr6ati  23775  fimacnvinrn2  23892  predidm  25213  osumcllem9N  30079  dihmeetbclemN  31420  dihmeetlem11N  31433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-in 3271
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