Consider the following premises. forall x_8: P_2(x_8) -> Q_8(x_8). forall x_8: P_2(x_8). forall x_8: ~Q_8(x_8). Can we infer exists y_2: R_2(y_2) from them? If possible, name the inference rules being used at each step.
 From forall x_8: P_2(x_8) -> Q_8(x_8), forall x_8: ~Q_8(x_8) we can infer forall x_8: ~P_2(x_8) via universal modus tollens. From forall x_8: P_2(x_8) we can infer forall x_8: P_2(x_8) or R_2(x_8) via universal addition. From forall x_8: P_2(x_8) or R_2(x_8), forall x_8: ~P_2(x_8) we can infer forall x_8: R_2(x_8) via universal disjunctive syllogism. From forall x_8: R_2(x_8) we can infer R_2(c) via universal instantiation. Finally, from R_2(c) we can infer exists y_2: R_2(y_2) via existential generalization. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.