The set of valid sentences in FO is not decidable as a consequence of rec. inseparability
The chosen finite axiomatization of $\mathbb N$ is consistent, since $\mathbb N$ is a model for them. Hence, any sentence provable from it must be satisfiable (e.g. by $\mathbb N$).

From this, we get that the set of provable sentences is disjoint from the set of unsatisfiable formulas.

Hence, the set of unsatisfiable formulas must be undecidable, otherwise we would have separability (taking $R$ to be the set of unsatisfiable formulas), and we would contradict the theorem.

Hence, the set of valid formulas must be undecidable as well, since we have the reduction "$P$ unsatisfiable $\iff \lnot P$ valid" (also mentioned by SimonJ above).