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Theorem isabl 15421
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 15420 . 2  |-  Abel  =  ( Grp  i^i CMnd )
21elin2 3533 1  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    e. wcel 1726   Grpcgrp 14690  CMndccmn 15417   Abelcabel 15418
This theorem is referenced by:  ablgrp  15422  ablcmn  15423  isabl2  15425  ablpropd  15427  isabld  15430  prdsabld  15482  unitabl  15778  tsmsinv  18182  tgptsmscls  18184  tsmsxplem1  18187  tsmsxplem2  18188  gicabl  27254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-abl 15420
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