Consider the following premises. forall y: ~T(y). forall y: U(y) -> T(y). forall y: U(y) -> U_1(y). forall y: U_1(y) -> U(y). forall y: W_1(y) <-> U_1(y). Can we infer exists y: W_1(y) from them? If possible, name the inference rules being used at each step.
 From forall y: U_1(y) -> U(y), forall y: U(y) -> U_1(y) we can infer forall y: U_1(y) <-> U(y) via universal biconditional introduction. From forall y: U(y) -> T(y), forall y: ~T(y) we can infer forall y: ~U(y) via universal modus tollens. From forall y: U_1(y) <-> U(y), forall y: ~U(y) we can infer forall y: ~U_1(y) via universal biconditional elimination. Finally, from forall y: W_1(y) <-> U_1(y), forall y: ~U_1(y) we can infer forall y: ~W_1(y) via universal biconditional elimination, which contradicts exists y: W_1(y). Therefore, the answer is no.