 \(\mathcal{M},s\models \widetilde{\exists }X\varphi\) if and only if \(\mathcal{M}^{\prime },s(\mathbf{X}/X)\models \varphi\) for some X-expansion \(\mathcal{M}^{\prime }\) of \(\mathcal{M}\) and some \(\mathbf{X}\in \wp (\mathbf{M}_{i_{1}}^{\prime }\times \ldots \times \mathbf{M}_{i_{r}}^{\prime })\) (where variable \(X\) is of sort \(\langle i_{1},\ldots ,i_{r},0\rangle)\)