Theorem gbowodd 48520

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 48517	. 2 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
2	1	simplbi 501	1 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ Odd )

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  (class class class)co 7410   + caddc 11098  ℙcprime 16724   Odd codd 48390   GoldbachOddW cgbow 48511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3090  df-rab 3417  df-v 3457  df-gbow 48514

This theorem is referenced by:  gboodd  48522