Theorem isgbe 48516

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 2767	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = (𝑝 + 𝑞) ↔ 𝑍 = (𝑝 + 𝑞)))
2	1	3anbi3d 1470	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞)) ↔ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))
3	2	2rexbidv 3230	. 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 48513	. 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 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  (class class class)co 7410   + caddc 11098  ℙcprime 16724   Even ceven 48389   Odd codd 48390   GoldbachEven cgbe 48510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3090  df-rab 3417  df-v 3457  df-gbe 48513

This theorem is referenced by:  gbeeven  48519  gbepos  48523  gbegt5  48526  6gbe  48536  8gbe  48538  sbgoldbwt  48542  sbgoldbst  48543  sbgoldbalt  48546  nnsum3primesgbe  48557  bgoldbtbndlem4  48573  bgoldbtbnd  48574