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Theorem caov12 6234
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 6203 . . 3  |-  ( A F B )  =  ( B F A )
54oveq1i 6050 . 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 6206 . 2  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
92, 1, 6, 7caovass 6206 . 2  |-  ( ( B F A ) F C )  =  ( B F ( A F C ) )
105, 8, 93eqtr3i 2432 1  |-  ( A F ( B F C ) )  =  ( B F ( A F C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   _Vcvv 2916  (class class class)co 6040
This theorem is referenced by:  caov31  6235  caov4  6237  caovmo  6243  distrnq  8794  ltaddnq  8807  ltexnq  8808  1idpr  8862  prlem934  8866  prlem936  8880  mulcmpblnrlem  8904  ltsosr  8925  0idsr  8928  1idsr  8929  recexsrlem  8934  mulgt0sr  8936  axmulass  8988
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043
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