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Theorem isgbow 47360
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 2730 . . . 4 (𝑧 = 𝑍 → (𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
21rexbidv 3169 . . 3 (𝑧 = 𝑍 → (∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
322rexbidv 3210 . 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 47357 . 2 GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
53, 4elrab2 3683 1 (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1534  wcel 2099  wrex 3060  (class class class)co 7416   + caddc 11152  cprime 16667   Odd codd 47233   GoldbachOddW cgbow 47354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rex 3061  df-rab 3420  df-v 3464  df-gbow 47357
This theorem is referenced by:  gbowodd  47363  gbogbow  47364  gbowpos  47367  gbowgt5  47370  gbowge7  47371  7gbow  47380  sbgoldbwt  47385  sbgoldbm  47392  nnsum4primesodd  47404
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