Consider the following premises. forall x_2: P_1(x_2) or ~U(x_2). exists x_2: U(x_2) or ~P_1(x_2). Can we infer exists x_2: P_1(x_2) -> U(x_2) from them?
 From forall x_2: P_1(x_2) or ~U(x_2) we can infer forall x_2: U(x_2) -> P_1(x_2). Finally, from exists x_2: U(x_2) or ~P_1(x_2) we can infer exists x_2: P_1(x_2) -> U(x_2). Therefore, the answer is yes.