Consider the following premises. exists y: U(y) -> P_3(y). forall y: U(y) -> W_2(y). forall y: W_2(y) -> U(y). forall y: ~P_3(y). Can we infer forall y: W_2(y) from them? If possible, name the inference rules being used at each step.
 From forall y: U(y) -> W_2(y), forall y: W_2(y) -> U(y) we can infer forall y: U(y) <-> W_2(y) via universal biconditional introduction. From exists y: U(y) -> P_3(y), forall y: ~P_3(y) we can infer exists y: ~U(y) via existential modus tollens. Finally, from forall y: U(y) <-> W_2(y), exists y: ~U(y) we can infer exists y: ~W_2(y) via existential biconditional elimination, which contradicts forall y: W_2(y). Therefore, the answer is no.