 By (As) we have \(\{\forall v\neg \theta_n (x|v),\theta_n\}\vdash \theta_n\) and by \((\forall\)E) we have \(\{\forall v\neg \theta_n (x|v), \theta_n\}\vdash \neg \theta_n\). So \(\{\forall v\neg \theta_n (x|v), \theta_n\}\) is inconsistent. Let \(\phi\) be any sentence of the language. By ex falso quodlibet (Theorem 10), we have that \(\{\forall v\neg \theta_n (x|v),\theta_n\}\vdash \phi\) and \(\{\forall v\neg \theta_n (x|v), \theta_n\}\vdash \neg \phi\). So with (2), we have that \(\Gamma_n, \forall v\neg \theta_n (x|v)\vdash \phi\) and \(\Gamma_n, \forall v\neg \theta_n (x|v)\vdash \neg \phi\), by \((\exists\)E). By Cut (Theorem 11), \(\Gamma_n \vdash \phi\) and \(\Gamma_n \vdash \neg \phi\). So \(\Gamma_n\) is inconsistent, contradicting the assumption. So \(\Gamma'\) is consistent.