Theorem ablgrpd 19700

Description: An Abelian group is a group, deduction form of ablgrp 19699. (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 19699	. 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2113  Grpcgrp 18848  Abelcabl 19695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-in 3905  df-abl 19697

This theorem is referenced by:  imasabl  19790  ablsimpgd  20032  rnggrp  20078  primrootscoprmpow  42213  primrootspoweq0  42220  aks6d1c6isolem1  42288  aks6d1c6isolem2  42289  aks6d1c6lem5  42291