Theorem gbeeven

Description: An even Goldbach number is even. (Contributed by AV, 25-Jul-2020)

	Ref	Expression
Assertion	gbeeven	⊢ ( 𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )

Proof

Step	Hyp	Ref	Expression
1		isgbe	⊢ ( 𝑍 ∈ GoldbachEven ↔ ( 𝑍 ∈ Even ∧ ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = ( 𝑝 + 𝑞 ) ) ) )
2	1	simplbi	⊢ ( 𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )