Consider the following premises. exists x_5: T_2(x_5) -> P_5(x_5). forall x_5: P_5(x_5) <-> T_2(x_5). forall x_5: ~T_2(x_5). Can we infer forall x_5: P_5(x_5) from them? If possible, name the inference rules being used at each step.
 From forall x_5: P_5(x_5) <-> T_2(x_5) we can infer forall x_5: P_5(x_5) -> T_2(x_5) via universal biconditional elimination. From exists x_5: T_2(x_5) -> P_5(x_5), forall x_5: P_5(x_5) -> T_2(x_5) we can infer exists x_5: T_2(x_5) <-> P_5(x_5) via existential biconditional introduction. Finally, from exists x_5: T_2(x_5) <-> P_5(x_5), forall x_5: ~T_2(x_5) we can infer exists x_5: ~P_5(x_5) via existential biconditional elimination, which contradicts forall x_5: P_5(x_5). Therefore, the answer is no.