Is Universality Theorem applicable to Halting problem?
By this book notation we have

$Halt(x,y)$ $\iff$ Program number $y$ eventually halts on input $x$.

Since there $Halt(x,y)$ is not computable predicate (theorem 2.1 page 68) there is no program that check if program number $y$ halts in input $x$ (for every $y$ and $x$), but this not mean that there is no program that run program number $y$ on input $x$!

By using universal program $\phi(x,y)$ we can run program number $y$ on input $x$. This means that $Halt(x,y)$ is r.e.

+if you think a bit more you can observe that program number y can get stuck in loop on input x, and that's why $Halt(x,y)$ is not computable.