Theorem gboodd 40177

Description: An odd Goldbach number is odd. (Contributed by AV, 25-Jul-2020.)

Assertion

Ref	Expression
gboodd	⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ Odd )

Proof of Theorem gboodd

Dummy variables 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isgbo 40174	. 2 ⊢ (𝑍 ∈ GoldbachOdd ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
2	1	simplbi 475	1 ⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ Odd )

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  (class class class)co 6549   + caddc 9818  ℙcprime 15223   Odd codd 40076   GoldbachOdd cgbo 40168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-gbo 40171

This theorem is referenced by:  gboaodd  40179