Consider the following premises. forall x_18: R_18(x_18) -> P(x_18). forall x_18: ~P_18(x_18) or R_18(x_18). forall x_18: S(x_18). exists x_18: Q_18(x_18) <-> P(x_18). forall x_18: P_18(x_18) or Q_18(x_18). Can we infer forall x_18: ~P(x_18) from them? If possible, name the inference rules being used at each step.
 From forall x_18: P_18(x_18) or Q_18(x_18), forall x_18: ~P_18(x_18) or R_18(x_18) we can infer forall x_18: Q_18(x_18) or R_18(x_18) via universal resolution. From exists x_18: Q_18(x_18) <-> P(x_18) we can infer exists x_18: Q_18(x_18) -> P(x_18) via existential biconditional elimination. From exists x_18: Q_18(x_18) -> P(x_18), forall x_18: R_18(x_18) -> P(x_18), forall x_18: Q_18(x_18) or R_18(x_18) we can infer exists x_18: P(x_18) via existential disjunction elimination. From exists x_18: P(x_18), forall x_18: S(x_18) we can infer exists x_18: P(x_18) and S(x_18) via existential conjunction. Finally, from exists x_18: P(x_18) and S(x_18) we can infer exists x_18: P(x_18) via existential simplification, which contradicts forall x_18: ~P(x_18). Therefore, the answer is no.