Consider the following premises. forall x_15: P_1(x_15) -> Q_7(x_15). exists x_15: P_15(x_15) <-> P_1(x_15). forall x_15: ~R_7(x_15). forall x_15: Q_7(x_15) -> R_7(x_15). forall x_15: P_15(x_15) -> P_1(x_15). Can we infer exists x_15: ~P_15(x_15) from them? If possible, name the inference rules being used at each step.
 From exists x_15: P_15(x_15) <-> P_1(x_15) we can infer exists x_15: P_1(x_15) -> P_15(x_15) via existential biconditional elimination. From forall x_15: P_1(x_15) -> Q_7(x_15), forall x_15: Q_7(x_15) -> R_7(x_15) we can infer forall x_15: P_1(x_15) -> R_7(x_15) via universal transitivity. From forall x_15: P_1(x_15) -> R_7(x_15), forall x_15: ~R_7(x_15) we can infer forall x_15: ~P_1(x_15) via universal modus tollens. From exists x_15: P_1(x_15) -> P_15(x_15), forall x_15: P_15(x_15) -> P_1(x_15) we can infer exists x_15: P_1(x_15) <-> P_15(x_15) via existential biconditional introduction. Finally, from exists x_15: P_1(x_15) <-> P_15(x_15), forall x_15: ~P_1(x_15) we can infer exists x_15: ~P_15(x_15) via existential biconditional elimination. Therefore, the answer is yes.