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Theorem isgbow 47677
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.
StepHypRef Expression
1 eqeq1 2739 . . . 4 (𝑧 = 𝑍 → (𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
21rexbidv 3177 . . 3 (𝑧 = 𝑍 → (∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
322rexbidv 3220 . 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 47674 . 2 GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
53, 4elrab2 3698 1 (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  (class class class)co 7431   + caddc 11156  cprime 16705   Odd codd 47550   GoldbachOddW cgbow 47671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-rab 3434  df-v 3480  df-gbow 47674
This theorem is referenced by:  gbowodd  47680  gbogbow  47681  gbowpos  47684  gbowgt5  47687  gbowge7  47688  7gbow  47697  sbgoldbwt  47702  sbgoldbm  47709  nnsum4primesodd  47721
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