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Theorem gbeeven 45916
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 45913 . 2 (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
21simplbi 498 1 (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  wrex 3072  (class class class)co 7354   + caddc 11051  cprime 16544   Even ceven 45786   Odd codd 45787   GoldbachEven cgbe 45907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rex 3073  df-rab 3407  df-v 3446  df-gbe 45910
This theorem is referenced by: (None)
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