Consider the following premises. forall x_15: P_15(x_15). exists x_15: P_15(x_15) -> Q_15(x_15). forall x_15: P_4(x_15) <-> T_1(x_15). forall x_15: Q_15(x_15) -> T_1(x_15). Can we infer forall x_15: ~P_4(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: P_15(x_15) we can infer exists x_15: Q_15(x_15) via existential modus ponens. From forall x_15: Q_15(x_15) -> T_1(x_15), exists x_15: Q_15(x_15) we can infer exists x_15: T_1(x_15) via existential modus ponens. From forall x_15: P_4(x_15) <-> T_1(x_15) we can infer forall x_15: T_1(x_15) -> P_4(x_15) via universal biconditional elimination. Finally, from forall x_15: T_1(x_15) -> P_4(x_15), exists x_15: T_1(x_15) we can infer exists x_15: P_4(x_15) via existential modus ponens, which contradicts forall x_15: ~P_4(x_15). Therefore, the answer is no.