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