Statements with multiple quantifiers Suppose $P(x,y)$ is a predicate whose truth depends on $x$ ($x\in D$) and $y$ ($y\in E$).
In the following statement,does the order of assigning values to $x$ and $y$ matter? For example, assign some value to $x$ and then find some appropriate value for $y$ (if it exists) or vice versa.
$\exists x$ in $D$ such that $\forall y$ in $E$: $P(x, y)$
 A: The statement
$$\exists x\in D\forall y\in E: P(x,y)$$
means that there exists some $x$ such that $P(x,y)$ is true for all $y$.
Therefore, to prove the statement, the only sensible "assigning of values" is:


*

*You pick some value for $x$ (you can decide which one)

*You pick an arbitrary value of $y$ (you can only assume that $y\in E$, you know nothing else about $y$.

*You prove that for this pair $x,y$, the statement $P(x,y)$ is true.



You ask: 

Can I first choose $y$ and then $x$?

This is equivalent to asking 

Is the statement $$\forall y\in E\exists x\in D: P(x,y)$$
  equivalent to the statement $$\exists x\in D\forall y\in E: P(x,y)$$

The answer is no.
For example, take $D=E=\mathbb N$ and $P(x,y) = "x> y"$
Then, the first statement, $$\forall y\in E\exists x\in D: P(x,y)$$ can be read as:

For all integers $y$, there exists some larger integer $x$

And this statement is obviously true. If $y$ is an integer, then $y+1$ is larger than $y$
On the other hand, the second statement, $$\exists x\in D\forall y\in E: P(x,y)$$, can be read as

There exists some $x$ so that $x$ is larger than any integer $y$.

This statement is not true.
A: 
In the following statement,does the order of assigning values to $x$ and $y$ matter? For example, assign some value to $x$ and then find some appropriate value for $y$ (if it exists) or vice versa.
$\exists x$ in $D$ such that $\forall y$ in $E$: $P(x, y)$

It may help to recast this statement as:
$\exists x:[x\in D \land \forall y:[y\in E \implies P(x,y)]]$
Then it should be obvious that you must assign (specify) a value tor the variable $x$ before you can assign one to $y$.
