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