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Theorem in13 3681
Description: A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
Assertion
Ref Expression
in13  |-  ( A  i^i  ( B  i^i  C ) )  =  ( C  i^i  ( B  i^i  A ) )

Proof of Theorem in13
StepHypRef Expression
1 in32 3680 . 2  |-  ( ( B  i^i  C )  i^i  A )  =  ( ( B  i^i  A )  i^i  C )
2 incom 3661 . 2  |-  ( A  i^i  ( B  i^i  C ) )  =  ( ( B  i^i  C
)  i^i  A )
3 incom 3661 . 2  |-  ( C  i^i  ( B  i^i  A ) )  =  ( ( B  i^i  A
)  i^i  C )
41, 2, 33eqtr4i 2468 1  |-  ( A  i^i  ( B  i^i  C ) )  =  ( C  i^i  ( B  i^i  A ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    i^i cin 3441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-in 3449
This theorem is referenced by:  inin  27985
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