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Theorem isgbe 40173
Description: The predicate "is an even Goldbach number". An even Goldbach number is an even number having a Goldbach partition, i.e. which can be written as sum of two odd primes. (Contributed by AV, 20-Jul-2020.)
Assertion
Ref Expression
isgbe (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
Distinct variable group:   𝑍,𝑝,𝑞

Proof of Theorem isgbe
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2614 . . . 4 (𝑧 = 𝑍 → (𝑧 = (𝑝 + 𝑞) ↔ 𝑍 = (𝑝 + 𝑞)))
213anbi3d 1397 . . 3 (𝑧 = 𝑍 → ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞)) ↔ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
322rexbidv 3039 . 2 (𝑧 = 𝑍 → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞)) ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
4 df-gbe 40170 . 2 GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
53, 4elrab2 3333 1 (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wrex 2897  (class class class)co 6549   + caddc 9818  cprime 15223   Even ceven 40075   Odd codd 40076   GoldbachEven cgbe 40167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-gbe 40170
This theorem is referenced by:  gbeeven  40176  gbepos  40180  gbegt5  40183  6gbe  40193  8gbe  40195  bgoldbwt  40199  bgoldbst  40200  sgoldbalt  40203  nnsum3primesgbe  40208  bgoldbtbndlem4  40224  bgoldbtbnd  40225
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