Consider the following premises. forall z_1: P_4(z_1) -> S_2(z_1). forall z_1: S_2(z_1). forall z_1: S_2(z_1) -> P_4(z_1). exists z_1: P(z_1). Can we infer exists z_1: P_4(z_1) from them?
 From forall z_1: P_4(z_1) -> S_2(z_1), forall z_1: S_2(z_1) -> P_4(z_1) we can infer forall z_1: P_4(z_1) <-> S_2(z_1). From forall z_1: P_4(z_1) <-> S_2(z_1) we can infer forall z_1: S_2(z_1) -> P_4(z_1). From forall z_1: S_2(z_1), exists z_1: P(z_1) we can infer exists z_1: S_2(z_1) and P(z_1). From exists z_1: S_2(z_1) and P(z_1) we can infer exists z_1: S_2(z_1). Finally, from forall z_1: S_2(z_1) -> P_4(z_1), exists z_1: S_2(z_1) we can infer exists z_1: P_4(z_1). Therefore, the answer is yes.