Theorem gbeeven 46422

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 46419	. 2 ⊢ (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
2	1	simplbi 499	1 ⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )

Syntax hints:   → wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  (class class class)co 7409   + caddc 11113  ℙcprime 16608   Even ceven 46292   Odd codd 46293   GoldbachEven cgbe 46413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3072  df-rab 3434  df-v 3477  df-gbe 46416

This theorem is referenced by: (None)