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Theorem gbeeven 41413
 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.
StepHypRef Expression
1 isgbe 41410 . 2 (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
21simplbi 476 1 (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1482   ∈ wcel 1989  ∃wrex 2912  (class class class)co 6647   + caddc 9936  ℙcprime 15379   Even ceven 41308   Odd codd 41309   GoldbachEven cgbe 41404 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rex 2917  df-rab 2920  df-v 3200  df-gbe 41407 This theorem is referenced by: (None)
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