Suppose you have a first order theory in some fixed language $L$. If the axiomatization only has an AE axiomatization, it would be considered $\Pi_2$ and if it is only EA it would be $\Sigma_2$.
Suppose that you have an axiomatization where some of the sentences are AE some others are EA. My question is what would you call the axiomatization? It is $\Pi_3$, but is it $\Pi_2$?
It is not $\Pi_2$. I've seen the notation "$\Pi_2\wedge\Sigma_2$" to describe such theories, although it's potentially ambiguous - it does not mean "both $\Pi_2$ and $\Sigma_2$".