Consider the following premises. forall x_15: P_11(x_15). forall x_15: T(x_15) -> ~S(x_15). forall x_15: Q_15(x_15) -> S(x_15). exists x_15: P_15(x_15) -> Q_15(x_15). forall x_15: P_11(x_15) -> P_15(x_15). Can we infer exists x_15: ~T(x_15) from them? If possible, name the inference rules being used at each step.
 From exists x_15: P_15(x_15) -> Q_15(x_15), forall x_15: Q_15(x_15) -> S(x_15) we can infer exists x_15: P_15(x_15) -> S(x_15) via existential transitivity. From forall x_15: P_11(x_15) -> P_15(x_15), exists x_15: P_15(x_15) -> S(x_15) we can infer exists x_15: P_11(x_15) -> S(x_15) via existential transitivity. From exists x_15: P_11(x_15) -> S(x_15), forall x_15: P_11(x_15) we can infer exists x_15: S(x_15) via existential modus ponens. Finally, from forall x_15: T(x_15) -> ~S(x_15), exists x_15: S(x_15) we can infer exists x_15: ~T(x_15) via existential modus tollens. Therefore, the answer is yes.