Theorem 4.5  (Parikh 1971)  Suppose that \(\text{I}\Delta_0 \vdash \forall \vec{x}  \exists y \phi(\vec{x},y)\) where \(\phi(\vec{x},y)\) is itself \(\Sigma_1\) . Then for some \(\mathcal{L}_a\)-term \(t(\vec{x})\), \(\text{I}\Delta_0 \vdash \forall \vec{x} \exists y \leq t(\vec{x}) \phi(\vec{x},y)\).