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 ) ) ) |
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 ) ) ) |