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Theorem caov12 6275
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
caov12  |-  ( A F ( B F C ) )  =  ( B F ( A F C ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    x, F, y, z

Proof of Theorem caov12
StepHypRef Expression
1 caov.1 . . . 4  |-  A  e. 
_V
2 caov.2 . . . 4  |-  B  e. 
_V
3 caov.com . . . 4  |-  ( x F y )  =  ( y F x )
41, 2, 3caovcom 6244 . . 3  |-  ( A F B )  =  ( B F A )
54oveq1i 6091 . 2  |-  ( ( A F B ) F C )  =  ( ( B F A ) F C )
6 caov.3 . . 3  |-  C  e. 
_V
7 caov.ass . . 3  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
81, 2, 6, 7caovass 6247 . 2  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
92, 1, 6, 7caovass 6247 . 2  |-  ( ( B F A ) F C )  =  ( B F ( A F C ) )
105, 8, 93eqtr3i 2464 1  |-  ( A F ( B F C ) )  =  ( B F ( A F C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2956  (class class class)co 6081
This theorem is referenced by:  caov31  6276  caov4  6278  caovmo  6284  distrnq  8838  ltaddnq  8851  ltexnq  8852  1idpr  8906  prlem934  8910  prlem936  8924  mulcmpblnrlem  8948  ltsosr  8969  0idsr  8972  1idsr  8973  recexsrlem  8978  mulgt0sr  8980  axmulass  9032
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084
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