Consider the following premises. forall x_17: P_3(x_17) <-> R(x_17). forall x_17: ~Q_17(x_17). forall x_17: R(x_17) <-> T(x_17). forall x_17: T(x_17) <-> T_2(x_17). exists x_17: T_2(x_17) -> Q_17(x_17). Can we infer exists x_17: ~P_3(x_17) from them? If possible, name the inference rules being used at each step.
 From exists x_17: T_2(x_17) -> Q_17(x_17), forall x_17: ~Q_17(x_17) we can infer exists x_17: ~T_2(x_17) via existential modus tollens. From forall x_17: T(x_17) <-> T_2(x_17), exists x_17: ~T_2(x_17) we can infer exists x_17: ~T(x_17) via existential biconditional elimination. From forall x_17: R(x_17) <-> T(x_17), exists x_17: ~T(x_17) we can infer exists x_17: ~R(x_17) via existential biconditional elimination. Finally, from forall x_17: P_3(x_17) <-> R(x_17), exists x_17: ~R(x_17) we can infer exists x_17: ~P_3(x_17) via existential biconditional elimination. Therefore, the answer is yes.