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Theorem inass 3511
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 3490 . . . . 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 3490 . . . 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 3490 . . 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 3494 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 1721    i^i cin 3279
This theorem is referenced by:  in12  3512  in32  3513  in4  3517  indif2  3544  difun1  3561  dfrab3ss  3579  dfif4  3710  onfr  4580  resres  5118  inres  5123  imainrect  5271  fresaun  5573  fresaunres2  5574  epfrs  7623  incexclem  12571  sadeq  12939  smuval2  12949  smumul  12960  ressinbas  13480  ressress  13481  resscatc  14215  sylow2a  15208  ablfac1eu  15586  ressmplbas2  16473  restco  17182  restopnb  17193  kgeni  17522  hausdiag  17630  fclsrest  18009  clsocv  19157  itg2cnlem2  19607  rplogsum  21174  chjassi  22941  pjoml2i  23040  cmcmlem  23046  cmbr3i  23055  fh1  23073  fh2  23074  pj3lem1  23662  dmdbr5  23764  mdslmd3i  23788  mdexchi  23791  atabsi  23857  dmdbr6ati  23879  fimacnvinrn2  24001  predidm  25402  osumcllem9N  30446  dihmeetbclemN  31787  dihmeetlem11N  31800
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-in 3287
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