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Theorem inass 3543
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 3522 . . . . 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 3522 . . . 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 3522 . . 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 3526 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 1652    e. wcel 1725    i^i cin 3311
This theorem is referenced by:  in12  3544  in32  3545  in4  3549  indif2  3576  difun1  3593  dfrab3ss  3611  dfif4  3742  onfr  4612  resres  5150  inres  5155  imainrect  5303  fresaun  5605  fresaunres2  5606  epfrs  7656  incexclem  12604  sadeq  12972  smuval2  12982  smumul  12993  ressinbas  13513  ressress  13514  resscatc  14248  sylow2a  15241  ablfac1eu  15619  ressmplbas2  16506  restco  17216  restopnb  17227  kgeni  17557  hausdiag  17665  fclsrest  18044  clsocv  19192  itg2cnlem2  19642  rplogsum  21209  chjassi  22976  pjoml2i  23075  cmcmlem  23081  cmbr3i  23090  fh1  23108  fh2  23109  pj3lem1  23697  dmdbr5  23799  mdslmd3i  23823  mdexchi  23826  atabsi  23892  dmdbr6ati  23914  fimacnvinrn2  24036  predidm  25443  osumcllem9N  30600  dihmeetbclemN  31941  dihmeetlem11N  31954
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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