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