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Theorem caovcom 5686
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 261 . 2  |-  ( A  e.  _V  /\  B  e.  _V )
4 caovcom.3 . . . 4  |-  ( x F y )  =  ( y F x )
54a1i 9 . . 3  |-  ( ( A  e.  _V  /\  ( x  e.  _V  /\  y  e.  _V )
)  ->  ( x F y )  =  ( y F x ) )
65caovcomg 5684 . 2  |-  ( ( A  e.  _V  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( A F B )  =  ( B F A ) )
71, 3, 6mp2an 410 1  |-  ( A F B )  =  ( B F A )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    = wceq 1259    e. wcel 1409   _Vcvv 2574  (class class class)co 5540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543
This theorem is referenced by:  caovord2  5701  caov32  5716  caov12  5717  ecopovsym  6233  ecopover  6235
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