Consider the following premises. forall x_17: P_17(x_17) <-> Q_17(x_17). forall x_17: ~Q_1(x_17). forall x_17: P_17(x_17) -> R_10(x_17). exists x_17: P_5(x_17) -> Q_17(x_17). forall x_17: R_10(x_17) -> Q_1(x_17). Can we infer forall x_17: P_5(x_17) from them? If possible, name the inference rules being used at each step.
 From forall x_17: P_17(x_17) <-> Q_17(x_17) we can infer forall x_17: Q_17(x_17) -> P_17(x_17) via universal biconditional elimination. From forall x_17: Q_17(x_17) -> P_17(x_17), forall x_17: P_17(x_17) -> R_10(x_17) we can infer forall x_17: Q_17(x_17) -> R_10(x_17) via universal transitivity. From exists x_17: P_5(x_17) -> Q_17(x_17), forall x_17: Q_17(x_17) -> R_10(x_17) we can infer exists x_17: P_5(x_17) -> R_10(x_17) via existential transitivity. From forall x_17: R_10(x_17) -> Q_1(x_17), forall x_17: ~Q_1(x_17) we can infer forall x_17: ~R_10(x_17) via universal modus tollens. Finally, from exists x_17: P_5(x_17) -> R_10(x_17), forall x_17: ~R_10(x_17) we can infer exists x_17: ~P_5(x_17) via existential modus tollens, which contradicts forall x_17: P_5(x_17). Therefore, the answer is no.