Consider the following premises. forall z_1: P_3(z_1) -> ~P_1(z_1). P_1(a_1). Can we infer forall z_1: P_3(z_1) from them? If possible, name the inference rules being used at each step.
 From P_1(a_1) we can infer exists z_1: P_1(z_1) via existential generalization. Finally, from forall z_1: P_3(z_1) -> ~P_1(z_1), exists z_1: P_1(z_1) we can infer exists z_1: ~P_3(z_1) via existential modus tollens, which contradicts forall z_1: P_3(z_1). Therefore, the answer is no.