Consider the following premises. exists x_12: P_10(x_12) -> P_12(x_12). forall x_12: P_12(x_12) -> P_10(x_12). forall x_12: T_3(x_12) or P_10(x_12). forall x_12: P_12(x_12) -> T_3(x_12). forall x_12: T_3(x_12) -> P_12(x_12). Can we infer exists x_12: P_12(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) -> T_3(x_12), forall x_12: T_3(x_12) -> P_12(x_12) we can infer forall x_12: P_12(x_12) <-> T_3(x_12) via universal biconditional introduction. From exists x_12: P_10(x_12) -> P_12(x_12), forall x_12: P_12(x_12) -> P_10(x_12) we can infer exists x_12: P_10(x_12) <-> P_12(x_12) via existential biconditional introduction. From forall x_12: P_12(x_12) <-> T_3(x_12) we can infer forall x_12: T_3(x_12) -> P_12(x_12) via universal biconditional elimination. From exists x_12: P_10(x_12) <-> P_12(x_12) we can infer exists x_12: P_10(x_12) -> P_12(x_12) via existential biconditional elimination. Finally, from forall x_12: T_3(x_12) -> P_12(x_12), exists x_12: P_10(x_12) -> P_12(x_12), forall x_12: T_3(x_12) or P_10(x_12) we can infer exists x_12: P_12(x_12) via existential disjunction elimination. Therefore, the answer is yes.