Consider the following premises. forall x_12: P_12(x_12) <-> P_3(x_12). exists x_12: P_12(x_12). Can we infer forall x_12: ~(P_3(x_12) or S(x_12)) from them?
 From forall x_12: P_12(x_12) <-> P_3(x_12) we can infer forall x_12: P_12(x_12) -> P_3(x_12). From forall x_12: P_12(x_12) -> P_3(x_12), exists x_12: P_12(x_12) we can infer exists x_12: P_3(x_12). Finally, from exists x_12: P_3(x_12) we can infer exists x_12: P_3(x_12) or S(x_12), which contradicts forall x_12: ~(P_3(x_12) or S(x_12)). Therefore, the answer is no.