How to read and interpret the mathematical 'for all statement' with an implication regarding sets? I'm fairly new to mathematical proofs and I was wondering how to interpret this example I came across in a textbook. Sadly, there are no answers so I don't know if i'm correct.

If $S$, $T$ and $U$ are three sets, then the statement $S \cap T \subseteq U$ can be written, using the logic symbols, as follows:
$(\forall x)[((x \in S)\wedge(x \in T))\implies(x ∈ U)]$

How would I interpret this in english?
I'm confused as to why the first part of this is $\forall x$ when it doesn't specify where $x$ is from but then proceeds to suggest that if it is an element of the set $S$ and $T$ then it means that it is a part of the set $U$.
And in set notation when we say $x \in S$ what is the $x$ in this case. Is it a representative of all the elements in the set S or is it suggesting that some other value of $x$ belongs to $S$. How should I go about thinking about this, when reading math?
 A: As has been mentioned in one of the comments, the english translation is "For all $x$, if $x$ is in both $S$ and $T$, then it is in $U$." This should agree with your intuition on what a subset is.
Your paragraph 

I'm confused as to why the first part of this is $\forall x$ when it doesn't specify where $x$ is from but then proceeds to suggest that if it is an element of the set $S$ and $T$ then it means that it is a part of the set $U$.

Is entirely correct. I'm guessing you are confused because we appear to be making a statement about elements of $S \cap T$. However, in this situation, we really are making a statement any object (in the universal set)---that's why you say "if it is an element of $S$..."
"$x \in S$" generally has two different usages. One is an assertion, as in "if $a,b$ are both integers, and $x = a +b$, then $x \in \mathbb{Z}$." Here, we are asserting that $x \in \mathbb{Z}$---it reads "$x$ is an integer."
The other common usage is (something similar to) "let $x \in S$..." In this case, $x$ is a representative of an element of $S$. 
The difference between the two is usually clear from context. After doing a few examples, and really understanding the basic set theory proofs, your brain just does it. 
A: There are some logical notations in which the $\forall$ symbol is followed by the name of a variable and the domain of the variable. 
In this notation, however, $x$ could be anything at all (or at least anything within some “universal” domain of objects).
And this is OK. Given any such $x$, if $x$ happens to be in the same domain from which the sets were selected, then the definition applies to $x$ in exactly the way you would expect. And if $x$ is not in the same domain, then it is not a member of any of the sets, and the definition applies to it the same way it applies to any other $x$ that is not a member of any of the three sets. 
