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Theorem ablpropd 15115
 Description: If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.)
Hypotheses
Ref Expression
ablpropd.1
ablpropd.2
ablpropd.3
Assertion
Ref Expression
ablpropd
Distinct variable groups:   ,,   ,,   ,,   ,,

Proof of Theorem ablpropd
StepHypRef Expression
1 ablpropd.1 . . . 4
2 ablpropd.2 . . . 4
3 ablpropd.3 . . . 4
41, 2, 3grppropd 14516 . . 3
51, 2, 3cmnpropd 15114 . . 3 CMnd CMnd
64, 5anbi12d 691 . 2 CMnd CMnd
7 isabl 15109 . 2 CMnd
8 isabl 15109 . 2 CMnd
96, 7, 83bitr4g 279 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1632   wcel 1696  cfv 5271  (class class class)co 5874  cbs 13164   cplusg 13224  cgrp 14378  CMndccmn 15105  cabel 15106 This theorem is referenced by:  ablprop  15116 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-0g 13420  df-mnd 14383  df-grp 14505  df-cmn 15107  df-abl 15108
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