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Theorem caov12 6280
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 6249 . . 3  |-  ( A F B )  =  ( B F A )
54oveq1i 6090 . 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 6252 . 2  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
92, 1, 6, 7caovass 6252 . 2  |-  ( ( B F A ) F C )  =  ( B F ( A F C ) )
105, 8, 93eqtr3i 2461 1  |-  ( A F ( B F C ) )  =  ( B F ( A F C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1362    e. wcel 1755   _Vcvv 2962  (class class class)co 6080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-iota 5369  df-fv 5414  df-ov 6083
This theorem is referenced by:  caov31  6281  caov4  6283  caovmo  6289  distrnq  9118  ltaddnq  9131  ltexnq  9132  1idpr  9186  prlem934  9190  prlem936  9204  mulcmpblnrlem  9228  ltsosr  9249  0idsr  9252  1idsr  9253  recexsrlem  9258  mulgt0sr  9260  axmulass  9312
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