Theorem gbeeven 48403

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 48400	. 2 ⊢ (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
2	1	simplbi 501	1 ⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  (class class class)co 7408   + caddc 11099  ℙcprime 16725   Even ceven 48273   Odd codd 48274   GoldbachEven cgbe 48394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rex 3096  df-rab 3424  df-v 3465  df-gbe 48397

This theorem is referenced by: (None)