 Strong inexpressibility of contradictory negation. As a corollary to the separation theorem, the result holds in a much stronger form. If \(\phi\) and \(\psi\) are sentences of \(\mathbf{IFL}\) such that \(M \vDash \phi\) iff \(M \not\vDash \psi\), then each of \(\phi\) and \(\psi\) is truth equivalent to a sentence of \(\mathbf{FO}\). Hence the contradictory negation \(\neg \phi\) is only expressible in \(\mathbf{IFL}\) for those \(\mathbf{IFL}\) sentences \(\phi\) that are truth equivalent to an \(\mathbf{FO}\)  sentence.[42]