ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caov13 Unicode version

Theorem caov13 5961
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
Hypotheses
Ref Expression
caov.1  |-  A  e. 
_V
caov.2  |-  B  e. 
_V
caov.3  |-  C  e. 
_V
caov.com  |-  ( x F y )  =  ( y F x )
caov.ass  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
Assertion
Ref Expression
caov13  |-  ( A F ( B F C ) )  =  ( C F ( B F A ) )
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    x, F, y, z

Proof of Theorem caov13
StepHypRef Expression
1 caov.1 . . 3  |-  A  e. 
_V
2 caov.2 . . 3  |-  B  e. 
_V
3 caov.3 . . 3  |-  C  e. 
_V
4 caov.com . . 3  |-  ( x F y )  =  ( y F x )
5 caov.ass . . 3  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
61, 2, 3, 4, 5caov31 5960 . 2  |-  ( ( A F B ) F C )  =  ( ( C F B ) F A )
71, 2, 3, 5caovass 5931 . 2  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
83, 2, 1, 5caovass 5931 . 2  |-  ( ( C F B ) F A )  =  ( C F ( B F A ) )
96, 7, 83eqtr3i 2168 1  |-  ( A F ( B F C ) )  =  ( C F ( B F A ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480   _Vcvv 2686  (class class class)co 5774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator