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Theorem gbowodd 46037
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 46034 . 2 (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
21simplbi 499 1 (𝑍 ∈ GoldbachOddW → 𝑍 ∈ Odd )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wrex 3070  (class class class)co 7361   + caddc 11062  cprime 16555   Odd codd 45907   GoldbachOddW cgbow 46028
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3071  df-rab 3407  df-v 3449  df-gbow 46031
This theorem is referenced by:  gboodd  46039
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