Subordination for sequentially equicontinuous equibounded $C_0$-semigroups

We consider operators $A$ on a sequentially complete Hausdorff locally convex space $X$ such that $-A$ generates a (sequentially) equicontinuous equibounded $C_0$-semigroup. For every Bernstein function $f$ we show that $-f(A)$ generates a semigroup which is of the same `kind' as the one generated by $-A$. As a special case we obtain that fractional powers $-A^{\alpha}$, where $\alpha \in (0,1)$, are generators.