Theorem gbeeven 47679

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 47676	. 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 1537   ∈ wcel 2106  ∃wrex 3068  (class class class)co 7431   + caddc 11156  ℙcprime 16705   Even ceven 47549   Odd codd 47550   GoldbachEven cgbe 47670

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-rab 3434  df-v 3480  df-gbe 47673

This theorem is referenced by: (None)