We give a new elementary proof of the comparison theorem relating $\sum^1_{n + 1}-\mathrm{AC}\uparrow$ and $\Pi^1_n -\mathrm{CA}\uparrow$; the proof does not use Skolem theories. By the same method we prove: a) $\sum^1_{n + 1}-\mathrm{DC} \uparrow \equiv (\Pi^1_n -CA)_{<\omega^\omega} \uparrow$, for suitable classes of sentences; b) $\sum^1_{n+1}-DC \uparrow$ proves the consistency of $(\Pi^1_n -CA)_\omega k$, for finite $k$, and hence is stronger than $\sum^1_{n+1}-AC \uparrow$. a) and b) answer a question of Feferman and Sieg.