Consider the following premises. exists x_12: P_5(x_12) <-> T(x_12). forall x_12: Q(x_12) or ~T(x_12). forall x_12: Q(x_12) -> T(x_12). Can we infer exists x_12: Q(x_12) -> P_5(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: Q(x_12) or ~T(x_12) we can infer forall x_12: T(x_12) -> Q(x_12) via universal rewriting. From forall x_12: T(x_12) -> Q(x_12), forall x_12: Q(x_12) -> T(x_12) we can infer forall x_12: T(x_12) <-> Q(x_12) via universal biconditional introduction. From forall x_12: T(x_12) <-> Q(x_12) we can infer forall x_12: Q(x_12) -> T(x_12) via universal biconditional elimination. From exists x_12: P_5(x_12) <-> T(x_12) we can infer exists x_12: T(x_12) -> P_5(x_12) via existential biconditional elimination. Finally, from forall x_12: Q(x_12) -> T(x_12), exists x_12: T(x_12) -> P_5(x_12) we can infer exists x_12: Q(x_12) -> P_5(x_12) via existential transitivity. Therefore, the answer is yes.