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Theorem in13 3716
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 3715 . 2  |-  ( ( B  i^i  C )  i^i  A )  =  ( ( B  i^i  A )  i^i  C )
2 incom 3696 . 2  |-  ( A  i^i  ( B  i^i  C ) )  =  ( ( B  i^i  C
)  i^i  A )
3 incom 3696 . 2  |-  ( C  i^i  ( B  i^i  A ) )  =  ( ( B  i^i  A
)  i^i  C )
41, 2, 33eqtr4i 2506 1  |-  ( A  i^i  ( B  i^i  C ) )  =  ( C  i^i  ( B  i^i  A ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    i^i cin 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-in 3488
This theorem is referenced by:  inin  27236
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