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Theorem in32 3381
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 3379 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
2 in12 3380 . 2  |-  ( A  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( A  i^i  C ) )
3 incom 3361 . 2  |-  ( B  i^i  ( A  i^i  C ) )  =  ( ( A  i^i  C
)  i^i  B )
41, 2, 33eqtri 2307 1  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    i^i cin 3151
This theorem is referenced by:  in13  3382  inrot  3384  wefrc  4387  imainrect  5119  fpwwe2  8265  incexclem  12295  ressress  13205  kgeni  17232  kgencn3  17253  fclsrest  17719  voliunlem1  18907  sspred  24174  selsubf  25990  selsubf3  25991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159
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