Theorem ablgrpd 19857

Description: An Abelian group is a group, deduction form of ablgrp 19856. (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 19856	. 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2143  Grpcgrp 19001  Abelcabl 19852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-in 3913  df-abl 19854

This theorem is referenced by:  imasabl  19947  ablsimpgd  20189  rnggrp  20237  primrootscoprmpow  42847  primrootspoweq0  42854  aks6d1c6isolem1  42922  aks6d1c6isolem2  42923  aks6d1c6lem5  42925