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Theorem caov31 6215
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
caov31  |-  ( ( 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 caov31
StepHypRef Expression
1 caov.1 . . . 4  |-  A  e. 
_V
2 caov.3 . . . 4  |-  C  e. 
_V
3 caov.2 . . . 4  |-  B  e. 
_V
4 caov.ass . . . 4  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
51, 2, 3, 4caovass 6186 . . 3  |-  ( ( A F C ) F B )  =  ( A F ( C F B ) )
6 caov.com . . . 4  |-  ( x F y )  =  ( y F x )
71, 2, 3, 6, 4caov12 6214 . . 3  |-  ( A F ( C F B ) )  =  ( C F ( A F B ) )
85, 7eqtri 2407 . 2  |-  ( ( A F C ) F B )  =  ( C F ( A F B ) )
91, 3, 2, 6, 4caov32 6213 . 2  |-  ( ( A F B ) F C )  =  ( ( A F C ) F B )
102, 1, 3, 6, 4caov32 6213 . . 3  |-  ( ( C F A ) F B )  =  ( ( C F B ) F A )
112, 1, 3, 4caovass 6186 . . 3  |-  ( ( C F A ) F B )  =  ( C F ( A F B ) )
1210, 11eqtr3i 2409 . 2  |-  ( ( C F B ) F A )  =  ( C F ( A F B ) )
138, 9, 123eqtr4i 2417 1  |-  ( ( A F B ) F C )  =  ( ( C F B ) F A )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   _Vcvv 2899  (class class class)co 6020
This theorem is referenced by:  caov13  6216  caov411  6218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-ov 6023
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