question about prime numbers

Question

Prove that for all odd prime numbers $p_1$ and $p_2$, there exist prime numbers(exclude 2) $p_3$ and $p_4$ such that $$p_3 + p_4 = p_1 + p_2 + 2.$$

Hints would be appreciated.

Answer 1

If this is true then the Goldbach conjecture is true. It is famous, and unproved.

Answer 2

For $p_1=2,p_2=7$, we have $p_1+p_2+2=11$, but there are no pairs of primes $(p_3,p_4)$ such that $$11=p_3+p_4.$$