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| Mirrors > Home > MPE Home > Th. List > Mathboxes > gbeeven | Structured version Visualization version GIF version | ||
| Description: An even Goldbach number is even. (Contributed by AV, 25-Jul-2020.) | 
| Ref | Expression | 
|---|---|
| gbeeven | ⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isgbe 47738 | . 2 ⊢ (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞)))) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 (class class class)co 7431 + caddc 11158 ℙcprime 16708 Even ceven 47611 Odd codd 47612 GoldbachEven cgbe 47732 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-rab 3437 df-v 3482 df-gbe 47735 | 
| This theorem is referenced by: (None) | 
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