 Suppose the last clause applied was \((\exists\mathrm{E})\). So we have \(\Gamma_1 \vdash \exists v\phi\) and \(\Gamma_2, \phi(v|t) \vdash \theta\), where \(\Gamma = \Gamma_1, \Gamma_2\), and \(t\) does not occur in \(\phi , \theta \), or in any member of \(\Gamma_2\).