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Theorem isabli 15428
 Description: Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)
Hypotheses
Ref Expression
isabli.g
isabli.b
isabli.p
isabli.c
Assertion
Ref Expression
isabli
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem isabli
StepHypRef Expression
1 isabli.g . 2
2 isabli.c . . 3
32rgen2a 2774 . 2
4 isabli.b . . 3
5 isabli.p . . 3
64, 5isabl2 15422 . 2
71, 3, 6mpbir2an 888 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  wral 2707  cfv 5456  (class class class)co 6083  cbs 13471   cplusg 13531  cgrp 14687  cabel 15415 This theorem is referenced by:  cnaddablx  15483  cnaddabl  15484  zaddablx  15485 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-grp 14814  df-cmn 15416  df-abl 15417
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