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Theorem caov12 6286
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 6255 . . 3  |-  ( A F B )  =  ( B F A )
54oveq1i 6096 . 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 6258 . 2  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
92, 1, 6, 7caovass 6258 . 2  |-  ( ( B F A ) F C )  =  ( B F ( A F C ) )
105, 8, 93eqtr3i 2466 1  |-  ( A F ( B F C ) )  =  ( B F ( A F C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   _Vcvv 2967  (class class class)co 6086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-iota 5376  df-fv 5421  df-ov 6089
This theorem is referenced by:  caov31  6287  caov4  6289  caovmo  6295  distrnq  9122  ltaddnq  9135  ltexnq  9136  1idpr  9190  prlem934  9194  prlem936  9208  mulcmpblnrlem  9232  ltsosr  9253  0idsr  9256  1idsr  9257  recexsrlem  9262  mulgt0sr  9264  axmulass  9316
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