MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  in31 Unicode version

Theorem in31 3396
Description: A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
Assertion
Ref Expression
in31  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( C  i^i  B )  i^i  A )

Proof of Theorem in31
StepHypRef Expression
1 in12 3393 . 2  |-  ( C  i^i  ( A  i^i  B ) )  =  ( A  i^i  ( C  i^i  B ) )
2 incom 3374 . 2  |-  ( ( A  i^i  B )  i^i  C )  =  ( C  i^i  ( A  i^i  B ) )
3 incom 3374 . 2  |-  ( ( C  i^i  B )  i^i  A )  =  ( A  i^i  ( C  i^i  B ) )
41, 2, 33eqtr4i 2326 1  |-  ( ( A  i^i  B )  i^i  C )  =  ( ( C  i^i  B )  i^i  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    i^i cin 3164
This theorem is referenced by:  inrot  3397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172
  Copyright terms: Public domain W3C validator