Theorem ablgrpd 19809

Description: An Abelian group is a group, deduction form of ablgrp 19808. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Hypothesis

Ref	Expression
ablgrpd.1	⊢ (𝜑 → 𝐺 ∈ Abel)

Assertion

Ref	Expression
ablgrpd	⊢ (𝜑 → 𝐺 ∈ Grp)

Proof of Theorem ablgrpd

Step	Hyp	Ref	Expression
1		ablgrpd.1	. 2 ⊢ (𝜑 → 𝐺 ∈ Abel)
2		ablgrp 19808	. 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2141  Grpcgrp 18958  Abelcabl 19804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-in 3911  df-abl 19806

This theorem is referenced by:  imasabl  19899  ablsimpgd  20141  rnggrp  20187  primrootscoprmpow  42680  primrootspoweq0  42687  aks6d1c6isolem1  42755  aks6d1c6isolem2  42756  aks6d1c6lem5  42758