 For the first half, assume that \(G_F\) were provable. Then, by the weak representability of provability-in-\(F\) by \(\Prov_F(x)\), \(F\) would also prove \(\Prov_F (\ulcorner G_F\urcorner)\).  However, because \(F\) in fact also proves the equivalence (G), i.e, \(F \vdash G_F \leftrightarrow \neg\Prov_F (\ulcorner G_F\urcorner)\), \(F\) would then prove \(\neg G_F\) too. But this would mean that \(F\) is inconsistent. In sum, if \(F\) is consistent, then \(G_F\) is not provable in \(F\). For this first half, the assumption of the simple consistency of \(F\) suffices.