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Theorem caovcom 5928
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 270 . 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 5926 . 2  |-  ( ( A  e.  _V  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  ( A F B )  =  ( B F A ) )
71, 3, 6mp2an 422 1  |-  ( A F B )  =  ( B F A )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2686  (class class class)co 5774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  caovord2  5943  caov32  5958  caov12  5959  ecopovsym  6525  ecopover  6527
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