Theorem gbowodd 47629

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 47626	. 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 2108  ∃wrex 3076  (class class class)co 7448   + caddc 11187  ℙcprime 16718   Odd codd 47499   GoldbachOddW cgbow 47620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rex 3077  df-rab 3444  df-v 3490  df-gbow 47623

This theorem is referenced by:  gboodd  47631