Is Universality Theorem applicable to Halting problem? This is Universality theorem In the Computability, Complexity and Languages book written by Davis in page 70:

If $\phi^{(n)}(x_1,...,x_n,y) = \psi_P(x_1,...,x_n)$        $where$  #$(P) = y$
Theorem 3.1 (Universality Theorem): for each $n&gt;0$ the function $\phi^{(n)}(x_1,...,x_n,y)$ is partially computable.

From what I understand of this theorem, the universal program is more like an interpreter it takes $x_1,...,x_n$ as input and $y$ as a program number and simulates what program with number $y$ does on the $x_i$s.
Since there is no program for halting problem so there is no $y$ for it to be passed to $\phi^{(n)}(x_1,...,x_n,y)$ so I think this theorem is not applicable to halting problem.Is it right?
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.