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Theorem ablgrpd 19703
Description: An Abelian group is a group, deduction form of ablgrp 19702. (Contributed by Rohan Ridenour, 3-Aug-2023.)
Hypothesis
Ref Expression
ablgrpd.1 (𝜑𝐺 ∈ Abel)
Assertion
Ref Expression
ablgrpd (𝜑𝐺 ∈ Grp)

Proof of Theorem ablgrpd
StepHypRef Expression
1 ablgrpd.1 . 2 (𝜑𝐺 ∈ Abel)
2 ablgrp 19702 . 2 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
31, 2syl 17 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Grpcgrp 18860  Abelcabl 19698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-abl 19700
This theorem is referenced by:  imasabl  19793  ablsimpgd  20035  rnggrp  20060  primrootscoprmpow  41478
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