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Theorem cmnpropd 17129
 Description: If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
ablpropd.1
ablpropd.2
ablpropd.3
Assertion
Ref Expression
cmnpropd CMnd CMnd
Distinct variable groups:   ,,   ,,   ,,   ,,

Proof of Theorem cmnpropd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablpropd.1 . . . 4
2 ablpropd.2 . . . 4
3 ablpropd.3 . . . 4
41, 2, 3mndpropd 16268 . . 3
53oveqrspc2v 6300 . . . . . 6
63oveqrspc2v 6300 . . . . . . 7
76ancom2s 803 . . . . . 6
85, 7eqeq12d 2424 . . . . 5
982ralbidva 2845 . . . 4
101raleqdv 3009 . . . . 5
111, 10raleqbidv 3017 . . . 4
122raleqdv 3009 . . . . 5
132, 12raleqbidv 3017 . . . 4
149, 11, 133bitr3d 283 . . 3
154, 14anbi12d 709 . 2
16 eqid 2402 . . 3
17 eqid 2402 . . 3
1816, 17iscmn 17127 . 2 CMnd
19 eqid 2402 . . 3
20 eqid 2402 . . 3
2119, 20iscmn 17127 . 2 CMnd
2215, 18, 213bitr4g 288 1 CMnd CMnd
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367   wceq 1405   wcel 1842  wral 2753  cfv 5568  (class class class)co 6277  cbs 14839   cplusg 14907  cmnd 16241  CMndccmn 17120 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524  ax-pow 4571 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-ov 6280  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-cmn 17122 This theorem is referenced by:  ablpropd  17130  crngpropd  17549  prdscrngd  17580  resvcmn  28267
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