Consider the following premises. forall x_22: P_22(x_22) or Q_22(x_22). forall x_22: Q_4(x_22) <-> T_4(x_22). exists x_22: ~P_22(x_22). forall x_22: Q_22(x_22) -> Q_14(x_22). forall x_22: Q_14(x_22) -> Q_4(x_22). Can we infer exists x_22: T_4(x_22) from them? If possible, name the inference rules being used at each step.
 From forall x_22: P_22(x_22) or Q_22(x_22), exists x_22: ~P_22(x_22) we can infer exists x_22: Q_22(x_22) via existential disjunctive syllogism. From forall x_22: Q_22(x_22) -> Q_14(x_22), forall x_22: Q_14(x_22) -> Q_4(x_22) we can infer forall x_22: Q_22(x_22) -> Q_4(x_22) via universal transitivity. From forall x_22: Q_4(x_22) <-> T_4(x_22) we can infer forall x_22: Q_4(x_22) -> T_4(x_22) via universal biconditional elimination. From forall x_22: Q_22(x_22) -> Q_4(x_22), exists x_22: Q_22(x_22) we can infer exists x_22: Q_4(x_22) via existential modus ponens. Finally, from forall x_22: Q_4(x_22) -> T_4(x_22), exists x_22: Q_4(x_22) we can infer exists x_22: T_4(x_22) via existential modus ponens. Therefore, the answer is yes.