Consider the following premises. P_1(a_1). forall z_1: S(z_1) -> R_3(z_1). forall z_1: S(z_1). Can we infer forall z_1: ~(P_1(z_1) and R_3(z_1)) from them?
 From forall z_1: S(z_1) -> R_3(z_1), forall z_1: S(z_1) we can infer forall z_1: R_3(z_1). From P_1(a_1) we can infer exists z_1: P_1(z_1). Finally, from exists z_1: P_1(z_1), forall z_1: R_3(z_1) we can infer exists z_1: P_1(z_1) and R_3(z_1), which contradicts forall z_1: ~(P_1(z_1) and R_3(z_1)). Therefore, the answer is no.