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Theorem in32 3650
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 3648 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
2 in12 3649 . 2  |-  ( A  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( A  i^i  C ) )
3 incom 3631 . 2  |-  ( B  i^i  ( A  i^i  C ) )  =  ( ( A  i^i  C
)  i^i  B )
41, 2, 33eqtri 2435 1  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    i^i cin 3412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-in 3420
This theorem is referenced by:  in13  3651  inrot  3653  wefrc  4816  imainrect  5265  sspred  5374  fpwwe2  9050  incexclem  13797  ressress  14904  kgeni  20328  kgencn3  20349  fclsrest  20815  voliunlem1  22250
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