Putting "$\forall y(y \in x \to \exists A \in F(y \in A))$" into words I'm new to mathematical proof and I struggle sometimes with putting definitions into words. If I had one like this:
$$\forall y(y \in x \to \exists A \in F(y \in A))$$
Would it be correct to read this as follows?

For all y such that if y is an element of x, then there exists a set A in a family of sets F such that y is an element of A.

 A: Compare:

*

*For each ball such that if it is green, then it is heavy.   
 (incoherent)

*For each ball such that if it is green, it is heavy.   
 (incoherent)

*For each ball, if it is green, then it is heavy.     (OK)

*For each ball, if it is green, it is heavy.     (OK)

*For each ball such that it is green, it is heavy.     (OK)

*For each ball that is green, it is heavy.     (better)

*For each green ball, it is heavy.     (even better)

*Every green ball is heavy.     (best)

Viewing the first bullet point as a chopped-off sentence, adding parentheses to clarify its structure, and completing it:

*

*For each ball such that (if it is green, then it is heavy), it is smooth.
  (now coherent)


$$\forall y\,\big(y \in x \to \exists A {\in} F\,(y \in A)\big)$$
Would it be correct to read this as follows?

For all $y$ such that if $y$ is an element of $x,$ then there exists a set $A$ in a family of sets $F$ such that y is an element of $A.$


As illustrated above, your suggested reading becomes grammatical once “if...then” or “such that” is dropped. Let's correct it by replacing “such that” with a comma:

*

*For all $y,$ if $y$ is an element of $x,$ then there exists a set $A$
in a family of sets $F$ such that $y$ is an element of $A.$
This literal reading can be condensed in several ways, the most succinctly as suggested by Pilcrow:

*

*Each element of $\color{red}x$ lies in some set in $\color{green}F.$
Abbreviating and rewriting the given formula: $$∀y{\in}\color{red}x\;∃A{\in}\color{green}F\;y\in A\\∀p{\in}\color{red}x\;∃q{\in}\color{green}F\;p\in q.$$
A: As far as I can see, “such that” here is completely incorrect, not just unclear. The symbolic statement claims that the implication is true for ALL $y$, whereas your statement instead defines (basically) a subset of all $y$ for which the implication is true (it’s very weird to do this without following it up with a statement about that set).
It might be easier to see with the following statements:
“For all $x$, if $x$ is an integer, then $1/x$ is an integer.”
“For all $x$ such that if $x$ is an integer, then $1/x$ is an integer...”
The first statement is just false, but the second statement is equivalent to “For all $x$ such that $x=1$ or $x$ is not an integer...”
Even if this had been logically correct, having an implication after a such that statement is generally grammatically unclear. You’re better off rephrasing it without one or the other. Following from the previous example, you could instead say
“For all $x$ such that $x$ is not an integer or $1/x$ is an integer...”
or even better
“If $x$ is not an integer or $1/x$ is an integer, then...”
