Representability theorem for computable functions

A [ logical theory] $T$ is said to satisfy the representability theorem for computable functions if for every [ computable function] $f:\mathbb{N}\mapsto\mathbb{N}$ there exists a two-place formula $\phi_f(x,y)$ in the [ language] of $T$ such that for all $n\in \mathbb{N}$ we have that $T\vdash \forall y\ \phi_f(\textbf n,y)\leftrightarrow y=\textbf{f(n)}$.