Addition is Arithmetically Definable

Theorem
Let $f: \N^2 \to \N$ be defined as:
 * $\map f {x, y} = x + y$

Then there exists a WFF $\map \phi {z, x, y}$ of $3$ free variables and containing no quantifiers such that:
 * $z = \map f {x, y} \iff \N \models \map \phi {\sqbrk z, \sqbrk x, \sqbrk y}$

where $\sqbrk a$ is the unary representation of $a \in \N$.

Proof
Define:
 * $\map \phi {z, x, y} := z = x + y$

Correctness is trivial.