Does existential elimination affect whether you can do a universal introduction? In exercise 1 of http://cnx.org/content/m10774/latest/, it says that you cannot do the universal introduction in

1   ∀y:(∃x:(R(x,y)))    Premise
2   ∃x:(R(x,q))         ∀Elim, line 1
3   R(p,q)              ∃Elim, line 2
4   ∀y:(R(p,y))         ∀Intro, line 3

because the existential elimination introduces the proxy variable p which could depend on the arbitrary variable q and hence q might no longer be arbitrary.
Does this make sense? If so, can you come up with examples of when this is the case?
 A: Take the universe to be the real numbers $R(x,y)$ to mean "$x\lt y$". Then the presmise says that for every real number $y$, there is a real number $x$ that is strictly smaller than $y$. This is true.
Line two eliminates the universal quantifier, and says that there is a real numbers $x$ which is smaller than $q$ ($q$ an unspecified, but fixed, real number). This is still true, regardless of what $q$ is. Say $q=1$.
Line 3 then eliminates the existential by selecting a specific $p$ which is smaller than $q$. This is still true; say, $p=0$.
Line 4 is now false: it would say "for every real number $y$, $0\lt y$. This is invalid.
The problem is that the $p$ introduced in line $3$ may depend on the $q$ introduced on line 2; it is not arbitrary and independent, so when you try introducing the universal quantifier in place of $q$, you are implicitly affecting $p$ as well.
A: We introduce a bit of romance.  Let $R(x,y)$ mean that $x$ loves $y$.  And let us assume, that as the old song says, everyone has someone who loves him/her (this is the first line).  Actually, we can be unromantic and not quite believe it, since we are merely deducing consequences from it.    
Take a particular individual $q$. The second line says that someone loves $q$.  This clearly follows from the first line.
Call one of the people who loves $q$ by the name $p$.  That gives us the third line.
Does $p$ love everybody?  (The fourth line asserts that (s)he does.)  Possible, but rather unlikely.  Anyway, it is certainly not deducible from the fact that $p$ loves $q$. 
Or if romance is not your style, what about biology?  Let $R(x,y)$ mean that $x$ is the mother of $y$.  The first line says that everybody has a mother. In the next few lines we try to deduce consequences. 
The second and third lines follow easily, as in the "loves" case.  So $p$ is the mother of $q$.  Surely we cannot deduce in the fourth line that $p$  is everyone's mother.  
