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Theorem caov31 6025
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 5996 . . 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 6024 . . 3  |-  ( A F ( C F B ) )  =  ( C F ( A F B ) )
85, 7eqtri 2185 . 2  |-  ( ( A F C ) F B )  =  ( C F ( A F B ) )
91, 3, 2, 6, 4caov32 6023 . 2  |-  ( ( A F B ) F C )  =  ( ( A F C ) F B )
102, 1, 3, 6, 4caov32 6023 . . 3  |-  ( ( C F A ) F B )  =  ( ( C F B ) F A )
112, 1, 3, 4caovass 5996 . . 3  |-  ( ( C F A ) F B )  =  ( C F ( A F B ) )
1210, 11eqtr3i 2187 . 2  |-  ( ( C F B ) F A )  =  ( C F ( A F B ) )
138, 9, 123eqtr4i 2195 1  |-  ( ( A F B ) F C )  =  ( ( C F B ) F A )
Colors of variables: wff set class
Syntax hints:    = wceq 1342    e. wcel 2135   _Vcvv 2724  (class class class)co 5839
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2726  df-un 3118  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-br 3980  df-iota 5150  df-fv 5193  df-ov 5842
This theorem is referenced by:  caov13  6026
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