Consider the following premises. exists x_5: T_2(x_5) -> W(x_5). forall x_5: ~Q_1(x_5). forall x_5: W(x_5) -> T_2(x_5). forall x_5: T_2(x_5) -> Q_1(x_5). Can we infer exists x_5: ~W(x_5) from them? If possible, name the inference rules being used at each step.
 From exists x_5: T_2(x_5) -> W(x_5), forall x_5: W(x_5) -> T_2(x_5) we can infer exists x_5: T_2(x_5) <-> W(x_5) via existential biconditional introduction. From forall x_5: T_2(x_5) -> Q_1(x_5), forall x_5: ~Q_1(x_5) we can infer forall x_5: ~T_2(x_5) via universal modus tollens. Finally, from exists x_5: T_2(x_5) <-> W(x_5), forall x_5: ~T_2(x_5) we can infer exists x_5: ~W(x_5) via existential biconditional elimination. Therefore, the answer is yes.