Confusion Between For Each and For All in Nested Quantifiers

Question

I am asked to translate the following into English:

$$\exists x \forall y \forall z\left((F(x,y) \wedge F(x,z) \wedge (y \neq z) \implies \neg F(y,z)\right)$$

The solution says:

This expression says that if students $x$ and $y$ are friends, and students $x$ and $z$ are friends, and furthermore, if $y$ and $z$ are not the same student, then $y$ and $z$ are not friends.

But I don't understand why $x$ and $y$ are for all and not existential quantifiers. Because it's not saying there is a student who is friends with all students -- just for one.

Answer 1

Given that $F(x,y)$ stands for "$x$ and $y$ are friends", and $St(x)$ stands for "$x$ is a student", the solution given is translation of the following formula:

$\forall x \forall y \forall z [St(x)\land St(y)\land St(z)\land F(x,y)\land F(x,z)\land y\neq z \rightarrow \neg F(y,z)]$

Answer 2

I would translate it as follows as the solution you quoted doesn't treat the quantifiers very explicitly:

There exists a student $x$ such that for any choice of students $y$ and $z$, where $y$ and $z$ are distinct, if $x$ is friends with both $y$ and $z$, then $y$ and $z$ are not friends.

Note that this is not saying $x$ is friends with every $y$ and $z$, but that if $x$ is friends with both, then $y$ and $z$ are not friends.