Theorem ablgrpd 19777

Description: An Abelian group is a group, deduction form of ablgrp 19776. (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 19776	. 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2107  Grpcgrp 18925  Abelcabl 19772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3466  df-in 3940  df-abl 19774

This theorem is referenced by:  imasabl  19867  ablsimpgd  20109  rnggrp  20128  primrootscoprmpow  42041  primrootspoweq0  42048  aks6d1c6isolem1  42116  aks6d1c6isolem2  42117  aks6d1c6lem5  42119