 That is, (1) for every \(\mathbf{IFL}[\tau]\) formula \(\phi\) there is an \(\mathbf{ESO}[\tau]\) formula \(\phi'\) such that for all \(\tau\) structures \(M\) and variable assignments \(g\), we have: \(M, g \vDash \phi\) iff \(M, g \vDash \phi'\). Actually, \(\mathrm{SK}[\phi]\) is a suitable \(\mathbf{ESO}\) formula. And (2) for every \(\mathbf{ESO}[\tau]\) formula \(\psi\) there is an \(\mathbf{IFL}[\tau]\) formula \(\psi'\) such that \(M, g \vDash \psi\) iff \(M, g \vDash \psi'\), for all \(\tau\) structures \(M\) and variable assignments \(g\). This follows from the fact that \(\mathbf{ESO}\) can be translated into \(\mathbf{FPO}\) (Enderton 1970, Walkoe 1970), which again can be translated into \(\mathbf{IFL}\).