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| Description: The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.) | 
| Ref | Expression | 
|---|---|
| isabl | ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-abl 19802 | . 2 ⊢ Abel = (Grp ∩ CMnd) | |
| 2 | 1 | elin2 4202 | 1 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2107 Grpcgrp 18952 CMndccmn 19799 Abelcabl 19800 | 
| 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 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-in 3957 df-abl 19802 | 
| This theorem is referenced by: ablgrp 19804 ablcmn 19806 isabl2 19809 ablpropd 19811 isabld 19814 ghmabl 19851 cntrabl 19862 prdsabld 19881 unitabl 20385 tsmsinv 24157 tgptsmscls 24159 tsmsxplem1 24162 tsmsxplem2 24163 abliso 33042 primrootsunit1 42099 gicabl 43116 2zrngaabl 48171 pgrpgt2nabl 48287 | 
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