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Theorem simpggrpd 19924
Description: A simple group is a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
Hypothesis
Ref Expression
simpggrpd.1 (𝜑𝐺 ∈ SimpGrp)
Assertion
Ref Expression
simpggrpd (𝜑𝐺 ∈ Grp)

Proof of Theorem simpggrpd
StepHypRef Expression
1 simpggrpd.1 . 2 (𝜑𝐺 ∈ SimpGrp)
2 simpggrp 19923 . 2 (𝐺 ∈ SimpGrp → 𝐺 ∈ Grp)
31, 2syl 17 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Grpcgrp 18794  SimpGrpcsimpg 19919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-iota 6484  df-fv 6540  df-simpg 19920
This theorem is referenced by:  simpgntrivd  19927  simpgnideld  19928  simpgnsgd  19929  ablsimpg1gend  19934  ablsimpgcygd  19935  ablsimpgfindlem1  19936  ablsimpgfindlem2  19937  ablsimpgfind  19939  ablsimpgprmd  19944  simpcntrab  45359
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