I need to prove that, given a computable binary tree $T$ whose paths are exactly the complete extensions of $PA$ (via some Gödel coding), there is a computable $X\subseteq\mathbb{N}$ such that for all $x\in X$ and $\sigma\in 2^x$ that can be extended to a path through $T$, both $\sigma0$ and $\sigma1$ can be extended to a path through $T$.
I know how to build such a tree, except that maybe I need to rearrange the coding to get this result. I believe that I should use Rosser's theorem at each stage $x\in X$, but I don't know how to use it for "different theories at one time", and infinitely many times. This is a homework problem.