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Theorem isabl 15186
Description: The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.)
Assertion
Ref Expression
isabl  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )

Proof of Theorem isabl
StepHypRef Expression
1 df-abl 15185 . 2  |-  Abel  =  ( Grp  i^i CMnd )
21elin2 3435 1  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1710   Grpcgrp 14455  CMndccmn 15182   Abelcabel 15183
This theorem is referenced by:  ablgrp  15187  ablcmn  15188  isabl2  15190  ablpropd  15192  isabld  15195  prdsabld  15247  unitabl  15543  tsmsinv  17926  tgptsmscls  17928  tsmsxplem1  17931  tsmsxplem2  17932  gicabl  26586
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866  df-in 3235  df-abl 15185
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