Consider the following premises. forall x_1: P_2(x_1). exists x_1: R(x_1). forall x_1: T_4(x_1) or ~R(x_1). Can we infer forall x_1: ~(P_2(x_1) and T_4(x_1)) from them? If possible, name the inference rules being used at each step.
 From forall x_1: T_4(x_1) or ~R(x_1) we can infer forall x_1: R(x_1) -> T_4(x_1) via universal rewriting. From forall x_1: R(x_1) -> T_4(x_1), exists x_1: R(x_1) we can infer exists x_1: T_4(x_1) via existential modus ponens. Finally, from forall x_1: P_2(x_1), exists x_1: T_4(x_1) we can infer exists x_1: P_2(x_1) and T_4(x_1) via existential conjunction, which contradicts forall x_1: ~(P_2(x_1) and T_4(x_1)). Therefore, the answer is no.