Consider the following premises. exists x: P_1(x) <-> P(x). forall x: U_1(x) or Q(x). forall x: ~U_1(x) or P(x). forall x: Q(x) -> P_1(x). Can we infer forall x: ~P_1(x) from them? If possible, name the inference rules being used at each step.
 From exists x: P_1(x) <-> P(x) we can infer exists x: P(x) -> P_1(x) via existential biconditional elimination. From forall x: U_1(x) or Q(x), forall x: ~U_1(x) or P(x) we can infer forall x: Q(x) or P(x) via universal resolution. Finally, from forall x: Q(x) -> P_1(x), exists x: P(x) -> P_1(x), forall x: Q(x) or P(x) we can infer exists x: P_1(x) via existential disjunction elimination, which contradicts forall x: ~P_1(x). Therefore, the answer is no.