Consider the following premises. forall x_1: ~R_1(x_1). exists x_1: R_2(x_1) -> Q_1(x_1). forall x_1: Q_1(x_1) -> R_1(x_1). Can we infer exists x_1: ~R_2(x_1) from them? If possible, name the inference rules being used at each step.
 From exists x_1: R_2(x_1) -> Q_1(x_1), forall x_1: Q_1(x_1) -> R_1(x_1) we can infer exists x_1: R_2(x_1) -> R_1(x_1) via existential transitivity. Finally, from exists x_1: R_2(x_1) -> R_1(x_1), forall x_1: ~R_1(x_1) we can infer exists x_1: ~R_2(x_1) via existential modus tollens. Therefore, the answer is yes.