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Metamath Proof Explorer
Description: If the strong binary Goldbach conjecture is valid, the binary Goldbach
conjecture is valid. (Contributed by AV, 23-Dec-2021)
|
|
Ref |
Expression |
|
Assertion |
sbgoldbb |
⊢ ( ∀ 𝑛 ∈ Even ( 4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀ 𝑛 ∈ Even ( 2 < 𝑛 → ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ 𝑛 = ( 𝑝 + 𝑞 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
sbgoldbalt |
⊢ ( ∀ 𝑛 ∈ Even ( 4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ ∀ 𝑛 ∈ Even ( 2 < 𝑛 → ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ 𝑛 = ( 𝑝 + 𝑞 ) ) ) |
| 2 |
1
|
biimpi |
⊢ ( ∀ 𝑛 ∈ Even ( 4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀ 𝑛 ∈ Even ( 2 < 𝑛 → ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ 𝑛 = ( 𝑝 + 𝑞 ) ) ) |