There are a number of patterns that are useful to remember, even though in any derivation they cannot be cited as formal rules:
(x)(y)(Fx & Gy) is equivalent to (x)Fx & (y)Gy.
(Ex)(Ey)(Fx v Gy) is equivalent to (Ex)Fx v (Ey)Gy.
(x)(y)(Fx -> Gy) is equivalent to (Ex)Fx -> (y)Gy.
(Ex)(Ey)(Fx -> Gy) is equivalent to (x)Fx -> (Ey)Gy.
The last two are a caution to be extremely careful not to confuse expressions such as (x)Fx->Ga and (x)(Fx->Ga).

You should practice deriving these patterns in order to develop a better sense for what is a legitimate derivation. Once we begin working with multiple quantifiers and two-place predicates, we no longer have a technique that will assure us ahead of time that any given formula is valid. The reason is that in order to observe the new name rule, we will find a consistency tree going on with open branches even in a valid pattern. (A technical expression is that our system is undecidable.) In order to see this for yourself, try a tree to test the equivalence of (x)(Ey)Fxy and (Ex)(y)Fyx, then do a simple proof by which you instantiate and then generalize.