Theorem isgbe 48373

Description: The predicate "is an even Goldbach number". An even Goldbach number is an even integer having a Goldbach partition, i.e. which can be written as a 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.

Step	Hyp	Ref	Expression
1		eqeq1 2766	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = (𝑝 + 𝑞) ↔ 𝑍 = (𝑝 + 𝑞)))
2	1	3anbi3d 1463	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞)) ↔ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
3	2	2rexbidv 3227	. 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 48370	. 2 ⊢ GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
5	3, 4	elrab2 3654	1 ⊢ (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∃wrex 3086  (class class class)co 7396   + caddc 11076  ℙcprime 16705   Even ceven 48246   Odd codd 48247   GoldbachEven cgbe 48367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1100  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3087  df-rab 3415  df-v 3456  df-gbe 48370

This theorem is referenced by:  gbeeven  48376  gbepos  48380  gbegt5  48383  6gbe  48393  8gbe  48395  sbgoldbwt  48399  sbgoldbst  48400  sbgoldbalt  48403  nnsum3primesgbe  48414  bgoldbtbndlem4  48430  bgoldbtbnd  48431