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Theorem gbeeven 44064
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 44061 . 2 (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
21simplbi 500 1 (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1083   = wceq 1537  wcel 2114  wrex 3126  (class class class)co 7133   + caddc 10518  cprime 15993   Even ceven 43934   Odd codd 43935   GoldbachEven cgbe 44055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rex 3131  df-rab 3134  df-v 3475  df-gbe 44058
This theorem is referenced by: (None)
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