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Theorem inass 3379
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 630 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  x  e.  C )  <->  ( x  e.  A  /\  (
x  e.  B  /\  x  e.  C )
) )
2 elin 3358 . . . . 5  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
32anbi2i 675 . . . 4  |-  ( ( x  e.  A  /\  x  e.  ( B  i^i  C ) )  <->  ( x  e.  A  /\  (
x  e.  B  /\  x  e.  C )
) )
41, 3bitr4i 243 . . 3  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  x  e.  C )  <->  ( x  e.  A  /\  x  e.  ( B  i^i  C
) ) )
5 elin 3358 . . . 4  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
65anbi1i 676 . . 3  |-  ( ( x  e.  ( A  i^i  B )  /\  x  e.  C )  <->  ( ( x  e.  A  /\  x  e.  B
)  /\  x  e.  C ) )
7 elin 3358 . . 3  |-  ( x  e.  ( A  i^i  ( B  i^i  C ) )  <->  ( x  e.  A  /\  x  e.  ( B  i^i  C
) ) )
84, 6, 73bitr4i 268 . 2  |-  ( ( x  e.  ( A  i^i  B )  /\  x  e.  C )  <->  x  e.  ( A  i^i  ( B  i^i  C ) ) )
98ineqri 3362 1  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151
This theorem is referenced by:  in12  3380  in32  3381  in4  3385  indif2  3412  difun1  3428  dfrab3ss  3446  dfif4  3576  onfr  4431  resres  4968  inres  4973  imainrect  5119  fresaun  5412  fresaunres2  5413  epfrs  7413  incexclem  12295  sadeq  12663  smuval2  12673  smumul  12684  ressinbas  13204  ressress  13205  resscatc  13937  sylow2a  14930  ablfac1eu  15308  ressmplbas2  16199  restco  16895  restopnb  16906  kgeni  17232  hausdiag  17339  fclsrest  17719  clsocv  18677  itg2cnlem2  19117  rplogsum  20676  chjassi  22065  pjoml2i  22164  cmcmlem  22170  cmbr3i  22179  fh1  22197  fh2  22198  pj3lem1  22786  dmdbr5  22888  mdslmd3i  22912  mdexchi  22915  atabsi  22981  dmdbr6ati  23003  fimacnvinrn2  23200  predidm  24188  islimrs3  25581  islimrs4  25582  hdrmp  25706  osumcllem9N  30153  dihmeetbclemN  31494  dihmeetlem11N  31507
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-in 3159
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