Consider the following premises. forall z: ~P_1(z). forall z: P_1(z) <-> R_1(z). forall z: W(z) -> R_1(z). Can we infer exists z: W(z) from them?
 From forall z: P_1(z) <-> R_1(z), forall z: ~P_1(z) we can infer forall z: ~R_1(z). Finally, from forall z: W(z) -> R_1(z), forall z: ~R_1(z) we can infer forall z: ~W(z), which contradicts exists z: W(z). Therefore, the answer is no.