Consider the following premises. forall x_8: P_8(x_8) <-> Q(x_8). forall x_8: ~Q_1(x_8). forall x_8: P_8(x_8) -> Q_1(x_8). Can we infer exists x_8: Q(x_8) from them? If possible, name the inference rules being used at each step.
 From forall x_8: P_8(x_8) <-> Q(x_8) we can infer forall x_8: Q(x_8) -> P_8(x_8) via universal biconditional elimination. From forall x_8: P_8(x_8) -> Q_1(x_8), forall x_8: ~Q_1(x_8) we can infer forall x_8: ~P_8(x_8) via universal modus tollens. Finally, from forall x_8: Q(x_8) -> P_8(x_8), forall x_8: ~P_8(x_8) we can infer forall x_8: ~Q(x_8) via universal modus tollens, which contradicts exists x_8: Q(x_8). Therefore, the answer is no.