Theorem isgbow 48243

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.)

Assertion

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

Distinct variable group:   𝑍,𝑝,𝑞,𝑟

Proof of Theorem isgbow

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2743	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
2	1	rexbidv 3163	. . 3 ⊢ (𝑧 = 𝑍 → (∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
3	2	2rexbidv 3204	. 2 ⊢ (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 48240	. 2 ⊢ GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
5	3, 4	elrab2 3632	1 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  (class class class)co 7356   + caddc 11032  ℙcprime 16631   Odd codd 48116   GoldbachOddW cgbow 48237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rex 3064  df-rab 3392  df-v 3433  df-gbow 48240

This theorem is referenced by:  gbowodd  48246  gbogbow  48247  gbowpos  48250  gbowgt5  48253  gbowge7  48254  7gbow  48263  sbgoldbwt  48268  sbgoldbm  48275  nnsum4primesodd  48287