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Theorem in13 3555
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 3554 . 2  |-  ( ( B  i^i  C )  i^i  A )  =  ( ( B  i^i  A )  i^i  C )
2 incom 3534 . 2  |-  ( A  i^i  ( B  i^i  C ) )  =  ( ( B  i^i  C
)  i^i  A )
3 incom 3534 . 2  |-  ( C  i^i  ( B  i^i  A ) )  =  ( ( B  i^i  A
)  i^i  C )
41, 2, 33eqtr4i 2467 1  |-  ( A  i^i  ( B  i^i  C ) )  =  ( C  i^i  ( B  i^i  A ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    i^i cin 3320
This theorem is referenced by:  inin  23997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-in 3328
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