Note that twin primes have form $6k\pm 1.$ If $6k+1$ is not prime then there exist integers $u$ and $v$ such that $6k+1=(6u+1)(6v+1)$ or $6k+1=(6u-1)(6v-1),$ implying that $k=6uv+u+v$ or $k=6uv-u-v.$ Similarly, if $6k-1$ is not prime then there exist integers $u$ and $v$ such that $6k+1=(6u+1)(6v-1)$, implying that $k=6uv-u+v.$ Therefore, if $k$ cannot be written in any of these forms then both $6k-1$ and $6k+1$ are primes.

That's only half of the question, no?

Change all if's into iff's