Metamath Proof Explorer

Theorem isgbow

Description: The predicate "is a weak odd Goldbach number". A weak odd Goldbach number is an odd integer having a Goldbach partition, i.e. which can be written as a sum of three primes. (Contributed by AV, 20-Jul-2020)

Ref Expression
Assertion isgbow ⊢ Z ∈ GoldbachOddW ↔ Z ∈ Odd ∧ ∃ p ∈ ℙ ∃ q ∈ ℙ ∃ r ∈ ℙ Z = p + q + r

Proof

Step Hyp Ref Expression
1 eqeq1 ⊢ z = Z → z = p + q + r ↔ Z = p + q + r
2 1 rexbidv ⊢ z = Z → ∃ r ∈ ℙ z = p + q + r ↔ ∃ r ∈ ℙ Z = p + q + r
3 2 2rexbidv ⊢ z = Z → ∃ p ∈ ℙ ∃ q ∈ ℙ ∃ r ∈ ℙ z = p + q + r ↔ ∃ p ∈ ℙ ∃ q ∈ ℙ ∃ r ∈ ℙ Z = p + q + r
4 df-gbow ⊢ GoldbachOddW = z ∈ Odd | ∃ p ∈ ℙ ∃ q ∈ ℙ ∃ r ∈ ℙ z = p + q + r
5 3 4 elrab2 ⊢ Z ∈ GoldbachOddW ↔ Z ∈ Odd ∧ ∃ p ∈ ℙ ∃ q ∈ ℙ ∃ r ∈ ℙ Z = p + q + r