Consider the following premises. forall x_2: P_2(x_2) or ~S(x_2). exists x_2: ~P_2(x_2). Can we infer exists x_2: ~S(x_2) from them?
 From forall x_2: P_2(x_2) or ~S(x_2) we can infer forall x_2: S(x_2) -> P_2(x_2). Finally, from forall x_2: S(x_2) -> P_2(x_2), exists x_2: ~P_2(x_2) we can infer exists x_2: ~S(x_2). Therefore, the answer is yes.