Consider the following premises. forall z: U_4(z) -> W(z). forall z: ~W(z). forall z: W(z) -> U_4(z). exists z: W(z) -> U_4(z). Can we infer forall z: U_4(z) from them? If possible, name the inference rules being used at each step.
 From forall z: U_4(z) -> W(z), forall z: W(z) -> U_4(z) we can infer forall z: U_4(z) <-> W(z) via universal biconditional introduction. From forall z: U_4(z) <-> W(z) we can infer forall z: U_4(z) -> W(z) via universal biconditional elimination. From exists z: W(z) -> U_4(z), forall z: U_4(z) -> W(z) we can infer exists z: W(z) <-> U_4(z) via existential biconditional introduction. From exists z: W(z) <-> U_4(z) we can infer exists z: U_4(z) -> W(z) via existential biconditional elimination. Finally, from exists z: U_4(z) -> W(z), forall z: ~W(z) we can infer exists z: ~U_4(z) via existential modus tollens, which contradicts forall z: U_4(z). Therefore, the answer is no.