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Theorem caovcom 6017
Description: Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)
Hypotheses
Ref Expression
caovcom.1  |-  A  e. 
_V
caovcom.2  |-  B  e. 
_V
caovcom.3  |-  ( x F y )  =  ( y F x )
Assertion
Ref Expression
caovcom  |-  ( A F B )  =  ( B F A )
Distinct variable groups:    x, y, A    x, B, y    x, F, y

Proof of Theorem caovcom
StepHypRef Expression
1 caovcom.1 . 2  |-  A  e. 
_V
2 caovcom.2 . . 3  |-  B  e. 
_V
31, 2pm3.2i 441 . 2  |-  ( A  e.  _V  /\  B  e.  _V )
4 caovcom.3 . . . 4  |-  ( x F y )  =  ( y F x )
54a1i 10 . . 3  |-  ( ( A  e.  _V  /\  ( x  e.  _V  /\  y  e.  _V )
)  ->  ( x F y )  =  ( y F x ) )
65caovcomg 6015 . 2  |-  ( ( A  e.  _V  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( A F B )  =  ( B F A ) )
71, 3, 6mp2an 653 1  |-  ( A F B )  =  ( B F A )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788  (class class class)co 5858
This theorem is referenced by:  caovord2  6032  caov32  6047  caov12  6048  caov42  6053  caovdir  6054  caovmo  6057  ecopovsym  6760  ecopover  6762  genpcl  8632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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