Consider the following premises. forall x_15: P_8(x_15) -> R_1(x_15). forall x_15: P_8(x_15) <-> S_2(x_15). forall x_15: P_12(x_15) -> S_2(x_15). forall x_15: P_12(x_15) or P_15(x_15). forall x_15: P_15(x_15) <-> S_2(x_15). Can we infer exists x_15: ~R_1(x_15) from them? If possible, name the inference rules being used at each step.
 From forall x_15: P_15(x_15) <-> S_2(x_15) we can infer forall x_15: P_15(x_15) -> S_2(x_15) via universal biconditional elimination. From forall x_15: P_12(x_15) -> S_2(x_15), forall x_15: P_15(x_15) -> S_2(x_15), forall x_15: P_12(x_15) or P_15(x_15) we can infer forall x_15: S_2(x_15) via universal disjunction elimination. From forall x_15: P_8(x_15) <-> S_2(x_15) we can infer forall x_15: S_2(x_15) -> P_8(x_15) via universal biconditional elimination. From forall x_15: S_2(x_15) -> P_8(x_15), forall x_15: P_8(x_15) -> R_1(x_15) we can infer forall x_15: S_2(x_15) -> R_1(x_15) via universal transitivity. Finally, from forall x_15: S_2(x_15) -> R_1(x_15), forall x_15: S_2(x_15) we can infer forall x_15: R_1(x_15) via universal modus ponens, which contradicts exists x_15: ~R_1(x_15). Therefore, the answer is no.