Theorem ablgrpd 19887

Description: An Abelian group is a group, deduction form of ablgrp 19886. (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 19886	. 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Grpcgrp 19031  Abelcabl 19882

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-in 3915  df-abl 19884

This theorem is used by:  imasabl  19977  ablsimpgd  20219  rnggrp  20267  primrootscoprmpow  42907  primrootspoweq0  42914  aks6d1c6isolem1  42982  aks6d1c6isolem2  42983  aks6d1c6lem5  42985