Theorem gbowodd 47680

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.

Step	Hyp	Ref	Expression
1		isgbow 47677	. 2 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
2	1	simplbi 497	1 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ Odd )

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  (class class class)co 7431   + caddc 11156  ℙcprime 16705   Odd codd 47550   GoldbachOddW cgbow 47671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-rab 3434  df-v 3480  df-gbow 47674

This theorem is referenced by:  gboodd  47682