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Theorem caov13 5716
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 5715 . 2  |-  ( ( A F B ) F C )  =  ( ( C F B ) F A )
71, 2, 3, 5caovass 5686 . 2  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
83, 2, 1, 5caovass 5686 . 2  |-  ( ( C F B ) F A )  =  ( C F ( B F A ) )
96, 7, 83eqtr3i 2110 1  |-  ( A F ( B F C ) )  =  ( C F ( B F A ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   _Vcvv 2602  (class class class)co 5537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-iota 4891  df-fv 4934  df-ov 5540
This theorem is referenced by: (None)
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