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Theorem gbowodd 45207
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 45204 . 2 (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
21simplbi 498 1 (𝑍 ∈ GoldbachOddW → 𝑍 ∈ Odd )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  wrex 3065  (class class class)co 7275   + caddc 10874  cprime 16376   Odd codd 45077   GoldbachOddW cgbow 45198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3070  df-rab 3073  df-v 3434  df-gbow 45201
This theorem is referenced by:  gboodd  45209
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