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Theorem caov13 6177
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 6176 . 2  |-  ( ( A F B ) F C )  =  ( ( C F B ) F A )
71, 2, 3, 5caovass 6147 . 2  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
83, 2, 1, 5caovass 6147 . 2  |-  ( ( C F B ) F A )  =  ( C F ( B F A ) )
96, 7, 83eqtr3i 2394 1  |-  ( A F ( B F C ) )  =  ( C F ( B F A ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1647    e. wcel 1715   _Vcvv 2873  (class class class)co 5981
This theorem is referenced by:  ltsonq  8740  mulcmpblnrlem  8842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-iota 5322  df-fv 5366  df-ov 5984
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