I want to express the fact that for all $x \in A$ that have the property that for all $y\in x$ $T(x,y)$ is true and there exists an $u \in B$ such that $P(y,u)$ is true AND for all $v\in C$, $Q(y,v)$ and $R(x,v)$ are also true, then $S(x)$ is true.
(Note that this statement is just a toy statement I have invented, because I was not sure I understood this transformation well, so I wanted to make it difficult, keeping with the idea, that if I got this difficult one right, that I probably understood how to do the transformations)
Formulated in a formal language of predicate logic, would this be
$$ \forall x ( x\in A \land \ \ \forall y (y\in x \rightarrow \ ( T(x,y) \land \ldots $$ $$ \ldots \land ( \exists u ( u \in B \land P(y,u) \land \forall v ( v\in C \rightarrow ( Q(y,v) \land R(x,v)))))))\rightarrow S(x) ) $$
(hope I didn't forget any paranthesis...) ?
Is there also a way to move the quantifiers "$\forall$" at the front, so that that string starts with $\forall x \forall y \ldots$ ?