Theorem gbeeven 43912

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

Assertion

Ref	Expression
gbeeven	⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )

Proof of Theorem gbeeven

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

Step	Hyp	Ref	Expression
1		isgbe 43909	. 2 ⊢ (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
2	1	simplbi 500	1 ⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  (class class class)co 7150   + caddc 10534  ℙcprime 16009   Even ceven 43782   Odd codd 43783   GoldbachEven cgbe 43903

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3497  df-gbe 43906

This theorem is referenced by: (None)