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Theorem in13 3382
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 3381 . 2  |-  ( ( B  i^i  C )  i^i  A )  =  ( ( B  i^i  A )  i^i  C )
2 incom 3361 . 2  |-  ( A  i^i  ( B  i^i  C ) )  =  ( ( B  i^i  C
)  i^i  A )
3 incom 3361 . 2  |-  ( C  i^i  ( B  i^i  A ) )  =  ( ( B  i^i  A
)  i^i  C )
41, 2, 33eqtr4i 2313 1  |-  ( A  i^i  ( B  i^i  C ) )  =  ( C  i^i  ( B  i^i  A ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    i^i cin 3151
This theorem is referenced by:  inin  23167
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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