  Akama et al. [1] introduced a hierarchical classification of first-order
formulas for a hierarchical prenex normal form theorem in semi-classical
arithmetic. In this paper, we give a justification for the hierarchical
classification in a general context of first-order theories. To this end, we
first formalize the standard transformation procedure for prenex normalization.
Then we show that the classes $\mathrm{E}_k$ and $\mathrm{U}_k$ introduced in
[1] are exactly the classes induced by $\Sigma_k$ and $\Pi_k$ respectively via
the transformation procedure in any first-order theory.
