$\forall {p_1\in\mathbb{P}, p_1>3},\ \exists {p_2\in\mathbb{P},\ p_3\in\mathbb{P}};\ (p_1 \neq p_2) \land (p_1\neq p_3) \land (p_1 = \frac{p_2+p_3}{2})$
Now I'm not a 100% sure about this, but I vaguely remember proving this once, but I cannot recall how I did it right now.
It's also a bit like a weaker version of Goldbach's conjecture, where now those even numbers that a double a prime are the sum of two primes, with the added condition of the summed primes being different.
So I'm asking if someone can provide/link to a proof of this? Because I've been looking around wikipedia and google but I cannot find this statement anywhere.