Yes, there is a way to mix them. The formula

is equivalent to

, both in the sense that they are true in the same models and in the sense that each one is derivable from the other. In fact, the latter formula is a prenex normal form of the former.

When we are talking about derivability, there is something similar to "existential elimination rule". It says that if

is already derived and from the assumption

with the free variable

one derives

, then

is derived (from the assumption

). In fact, this is nothing but formalization of a common way we deal with an assumption that asserts existence of something. So, having

and

, we apply existential elimination twice to get

with free variables

. After that, we introduce, i.e., add, existential quantifiers back.

If your doubts are about something else, feel free to clarify.