 By ex falso quodlibet (Theorem 10), \(\Gamma_n, \neg \exists x\theta_n, \exists x\theta_n \vdash \theta_n (x|c_i)\). So by \((\rightarrow\)I), \(\Gamma_n, \neg \exists x\theta_n \vdash(\exists x\theta_n \rightarrow \theta_n (x|c_i))\). From this and (1), we have \(\Gamma_n \vdash \neg \neg \exists x\theta_n\), by \((\neg\)I), and by (DNE) we have