Theorem isgbe 45447

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 2740	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = (𝑝 + 𝑞) ↔ 𝑍 = (𝑝 + 𝑞)))
2	1	3anbi3d 1442	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞)) ↔ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
3	2	2rexbidv 3210	. 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 45444	. 2 ⊢ GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
5	3, 4	elrab2 3632	1 ⊢ (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))

Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  ∃wrex 3071  (class class class)co 7307   + caddc 10924  ℙcprime 16425   Even ceven 45320   Odd codd 45321   GoldbachEven cgbe 45441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1089  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rex 3072  df-rab 3306  df-v 3439  df-gbe 45444

This theorem is referenced by:  gbeeven  45450  gbepos  45454  gbegt5  45457  6gbe  45467  8gbe  45469  sbgoldbwt  45473  sbgoldbst  45474  sbgoldbalt  45477  nnsum3primesgbe  45488  bgoldbtbndlem4  45504  bgoldbtbnd  45505