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Theorem in32 3553
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 3551 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
2 in12 3552 . 2  |-  ( A  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( A  i^i  C ) )
3 incom 3533 . 2  |-  ( B  i^i  ( A  i^i  C ) )  =  ( ( A  i^i  C
)  i^i  B )
41, 2, 33eqtri 2460 1  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    i^i cin 3319
This theorem is referenced by:  in13  3554  inrot  3556  wefrc  4576  imainrect  5312  fpwwe2  8518  incexclem  12616  ressress  13526  kgeni  17569  kgencn3  17590  fclsrest  18056  voliunlem1  19444  sspred  25447
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-in 3327
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