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Theorem caov12 6402
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 6371 . . 3  |-  ( A F B )  =  ( B F A )
54oveq1i 6211 . 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 6374 . 2  |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
92, 1, 6, 7caovass 6374 . 2  |-  ( ( B F A ) F C )  =  ( B F ( A F C ) )
105, 8, 93eqtr3i 2491 1  |-  ( A F ( B F C ) )  =  ( B F ( A F C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   _Vcvv 3078  (class class class)co 6201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-iota 5490  df-fv 5535  df-ov 6204
This theorem is referenced by:  caov31  6403  caov4  6405  caovmo  6411  distrnq  9242  ltaddnq  9255  ltexnq  9256  1idpr  9310  prlem934  9314  prlem936  9328  mulcmpblnrlem  9352  ltsosr  9373  0idsr  9376  1idsr  9377  recexsrlem  9382  mulgt0sr  9384  axmulass  9436
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