Consider the following premises. forall x_10: P_10(x_10) -> R_9(x_10). forall x_10: P(x_10) -> ~R_9(x_10). forall x_10: Q_10(x_10) -> R_9(x_10). exists x_10: P_10(x_10). Can we infer forall x_10: P(x_10) from them? If possible, name the inference rules being used at each step.
 From exists x_10: P_10(x_10) we can infer exists x_10: P_10(x_10) or Q_10(x_10) via existential addition. From forall x_10: P_10(x_10) -> R_9(x_10), forall x_10: Q_10(x_10) -> R_9(x_10), exists x_10: P_10(x_10) or Q_10(x_10) we can infer exists x_10: R_9(x_10) via existential disjunction elimination. Finally, from forall x_10: P(x_10) -> ~R_9(x_10), exists x_10: R_9(x_10) we can infer exists x_10: ~P(x_10) via existential modus tollens, which contradicts forall x_10: P(x_10). Therefore, the answer is no.