Theorem isabl 18197

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

Step	Hyp	Ref	Expression
1		df-abl 18196	. 2 |- Abel = ( Grp i^i CMnd )
2	1	elin2 3801	1 |- ( G e. Abel <-> ( G e. Grp /\ G e. CMnd ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   Grpcgrp 17422  CMndccmn 18193   Abelcabl 18194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-abl 18196

This theorem is referenced by:  ablgrp  18198  ablcmn  18199  isabl2  18201  ablpropd  18203  isabld  18206  ghmabl  18238  prdsabld  18265  unitabl  18668  tsmsinv  21951  tgptsmscls  21953  tsmsxplem1  21956  tsmsxplem2  21957  abliso  29696  gicabl  37669  2zrngaabl  41944  pgrpgt2nabl  42147