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Theorem in32 3671
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 3669 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
2 in12 3670 . 2  |-  ( A  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( A  i^i  C ) )
3 incom 3652 . 2  |-  ( B  i^i  ( A  i^i  C ) )  =  ( ( A  i^i  C
)  i^i  B )
41, 2, 33eqtri 2487 1  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    i^i cin 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-in 3444
This theorem is referenced by:  in13  3672  inrot  3674  wefrc  4823  imainrect  5388  fpwwe2  8922  incexclem  13418  ressress  14355  kgeni  19243  kgencn3  19264  fclsrest  19730  voliunlem1  21165  sspred  27778
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