Theorem isgbow 48401

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 2773	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
2	1	rexbidv 3195	. . 3 ⊢ (𝑧 = 𝑍 → (∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
3	2	2rexbidv 3236	. 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 48398	. 2 ⊢ GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
5	3, 4	elrab2 3663	1 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  (class class class)co 7408   + caddc 11099  ℙcprime 16725   Odd codd 48274   GoldbachOddW cgbow 48395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rex 3096  df-rab 3424  df-v 3465  df-gbow 48398

This theorem is referenced by:  gbowodd  48404  gbogbow  48405  gbowpos  48408  gbowgt5  48411  gbowge7  48412  7gbow  48421  sbgoldbwt  48426  sbgoldbm  48433  nnsum4primesodd  48445