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 ) ) )