 By (As), \(\Gamma_n, \theta_n (x|c_i), \exists x\theta_n \vdash \theta_n (x|c_i)\). So by \((\rightarrow\)I), \(\Gamma_n, \theta_n (x|c_i)\vdash(\exists x\theta_{n} \rightarrow \theta_{n}(x|c_i))\). From this and (1), we have \(\Gamma_n \vdash \neg \theta_n (x|c_i)\), by \((\neg\)I). Let \(t\) be a term that does not occur in \(\theta_n\) or in any member of \(\Gamma_n\). By uniform substitution of \(t\) for \(c_i\), we can turn the derivation of \(\Gamma_n \vdash \neg \theta_n (x|c_i)\) into \(\Gamma_n \vdash \neg \theta_n (x|t)\). By \((\forall\)I), we have