Theorem gbeeven 47735

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 47732	. 2 ⊢ (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
2	1	simplbi 497	1 ⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  (class class class)co 7410   + caddc 11137  ℙcprime 16695   Even ceven 47605   Odd codd 47606   GoldbachEven cgbe 47726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rex 3062  df-rab 3421  df-v 3466  df-gbe 47729

This theorem is referenced by: (None)