Consider the following premises. exists x: U(x) -> P_1(x). forall x: U(x) or ~Q(x). Can we infer exists x: Q(x) -> P_1(x) from them? If possible, name the inference rules being used at each step.
 From forall x: U(x) or ~Q(x) we can infer forall x: Q(x) -> U(x) via universal rewriting. Finally, from forall x: Q(x) -> U(x), exists x: U(x) -> P_1(x) we can infer exists x: Q(x) -> P_1(x) via existential transitivity. Therefore, the answer is yes.