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Theorem isabl 15093
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 15092 . 2  |-  Abel  =  ( Grp  i^i CMnd )
21elin2 3359 1  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   Grpcgrp 14362  CMndccmn 15089   Abelcabel 15090
This theorem is referenced by:  ablgrp  15094  ablcmn  15095  isabl2  15097  ablpropd  15099  isabld  15102  prdsabld  15154  unitabl  15450  tsmsinv  17830  tgptsmscls  17832  tsmsxplem1  17835  tsmsxplem2  17836  gicabl  27263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-abl 15092
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