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Theorem gbowodd 43927
 Description: A weak odd Goldbach number is odd. (Contributed by AV, 25-Jul-2020.)
Assertion
Ref Expression
gbowodd (𝑍 ∈ GoldbachOddW → 𝑍 ∈ Odd )

Proof of Theorem gbowodd
Dummy variables 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isgbow 43924 . 2 (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
21simplbi 500 1 (𝑍 ∈ GoldbachOddW → 𝑍 ∈ Odd )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  (class class class)co 7158   + caddc 10542  ℙcprime 16017   Odd codd 43797   GoldbachOddW cgbow 43918 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rex 3146  df-rab 3149  df-v 3498  df-gbow 43921 This theorem is referenced by:  gboodd  43929
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