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Theorem in32 3415
Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
in32  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i  B )

Proof of Theorem in32
StepHypRef Expression
1 inass 3413 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
2 in12 3414 . 2  |-  ( A  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( A  i^i  C ) )
3 incom 3395 . 2  |-  ( B  i^i  ( A  i^i  C ) )  =  ( ( A  i^i  C
)  i^i  B )
41, 2, 33eqtri 2340 1  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1633    i^i cin 3185
This theorem is referenced by:  in13  3416  inrot  3418  wefrc  4424  imainrect  5156  fpwwe2  8310  incexclem  12342  ressress  13252  kgeni  17288  kgencn3  17309  fclsrest  17771  voliunlem1  18960  sspred  24559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-in 3193
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