Consider the following premises. forall z: ~W_4(z). forall z: W_4(z) <-> P_4(z). forall z: S(z) <-> P_4(z). Can we infer exists z: S(z) from them? If possible, name the inference rules being used at each step.
 From forall z: W_4(z) <-> P_4(z), forall z: ~W_4(z) we can infer forall z: ~P_4(z) via universal biconditional elimination. Finally, from forall z: S(z) <-> P_4(z), forall z: ~P_4(z) we can infer forall z: ~S(z) via universal biconditional elimination, which contradicts exists z: S(z). Therefore, the answer is no.