Theorem isgbow 44820

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 2740	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
2	1	rexbidv 3206	. . 3 ⊢ (𝑧 = 𝑍 → (∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
3	2	2rexbidv 3209	. 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 44817	. 2 ⊢ GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
5	3, 4	elrab2 3594	1 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∃wrex 3052  (class class class)co 7191   + caddc 10697  ℙcprime 16191   Odd codd 44693   GoldbachOddW cgbow 44814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-rex 3057  df-rab 3060  df-v 3400  df-gbow 44817

This theorem is referenced by:  gbowodd  44823  gbogbow  44824  gbowpos  44827  gbowgt5  44830  gbowge7  44831  7gbow  44840  sbgoldbwt  44845  sbgoldbm  44852  nnsum4primesodd  44864