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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isgbe | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
isgbe | ⊢ (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2730 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = (𝑝 + 𝑞) ↔ 𝑍 = (𝑝 + 𝑞))) | |
2 | 1 | 3anbi3d 1439 | . . 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 47320 | . 2 ⊢ GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))} | |
5 | 3, 4 | elrab2 3684 | 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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∃wrex 3060 (class class class)co 7424 + caddc 11161 ℙcprime 16672 Even ceven 47196 Odd codd 47197 GoldbachEven cgbe 47317 |
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-3an 1086 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-gbe 47320 |
This theorem is referenced by: gbeeven 47326 gbepos 47330 gbegt5 47333 6gbe 47343 8gbe 47345 sbgoldbwt 47349 sbgoldbst 47350 sbgoldbalt 47353 nnsum3primesgbe 47364 bgoldbtbndlem4 47380 bgoldbtbnd 47381 |
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