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Theorem isabl 15371
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 15370 . 2  |-  Abel  =  ( Grp  i^i CMnd )
21elin2 3491 1  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1721   Grpcgrp 14640  CMndccmn 15367   Abelcabel 15368
This theorem is referenced by:  ablgrp  15372  ablcmn  15373  isabl2  15375  ablpropd  15377  isabld  15380  prdsabld  15432  unitabl  15728  tsmsinv  18130  tgptsmscls  18132  tsmsxplem1  18135  tsmsxplem2  18136  gicabl  27131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-in 3287  df-abl 15370
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