| Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
| 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 2756 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 = (𝑝 + 𝑞) ↔ 𝑍 = (𝑝 + 𝑞))) | |
| 2 | 1 | 3anbi3d 1453 | . . 3 ⊢ (𝑧 = 𝑍 → ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞)) ↔ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞)))) |
| 3 | 2 | 2rexbidv 3217 | . 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 48308 | . 2 ⊢ GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))} | |
| 5 | 3, 4 | elrab2 3644 | 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 208 ∧ wa 398 ∧ w3a 1095 = wceq 1550 ∈ wcel 2132 ∃wrex 3076 (class class class)co 7381 + caddc 11062 ℙcprime 16677 Even ceven 48184 Odd codd 48185 GoldbachEven cgbe 48305 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-ext 2724 |
| This theorem depends on definitions: df-bi 209 df-an 399 df-3an 1097 df-tru 1553 df-ex 1790 df-sb 2081 df-clab 2731 df-cleq 2744 df-clel 2827 df-rex 3077 df-rab 3405 df-v 3446 df-gbe 48308 |
| This theorem is referenced by: gbeeven 48314 gbepos 48318 gbegt5 48321 6gbe 48331 8gbe 48333 sbgoldbwt 48337 sbgoldbst 48338 sbgoldbalt 48341 nnsum3primesgbe 48352 bgoldbtbndlem4 48368 bgoldbtbnd 48369 |
| Copyright terms: Public domain | W3C validator |