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Theorem isabl2 15425
 Description: The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
iscmn.b
iscmn.p
Assertion
Ref Expression
isabl2
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem isabl2
StepHypRef Expression
1 isabl 15421 . 2 CMnd
2 grpmnd 14822 . . . 4
3 iscmn.b . . . . . 6
4 iscmn.p . . . . . 6
53, 4iscmn 15424 . . . . 5 CMnd
65baib 873 . . . 4 CMnd
72, 6syl 16 . . 3 CMnd
87pm5.32i 620 . 2 CMnd
91, 8bitri 242 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1653   wcel 1726  wral 2707  cfv 5457  (class class class)co 6084  cbs 13474   cplusg 13534  cmnd 14689  cgrp 14690  CMndccmn 15417  cabel 15418 This theorem is referenced by:  isabli  15431  invghm  15458  divsabl  15485 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-3an 939  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-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087  df-grp 14817  df-cmn 15419  df-abl 15420
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