Consider the following premises. forall y: T(y) or ~U(y). forall y: S_2(y) -> T(y). forall y: T(y) -> U(y). Can we infer exists y: ~(S_2(y) -> U(y)) from them? If possible, name the inference rules being used at each step.
 From forall y: T(y) or ~U(y) we can infer forall y: U(y) -> T(y) via universal rewriting. From forall y: U(y) -> T(y), forall y: T(y) -> U(y) we can infer forall y: U(y) <-> T(y) via universal biconditional introduction. From forall y: U(y) <-> T(y) we can infer forall y: T(y) -> U(y) via universal biconditional elimination. Finally, from forall y: S_2(y) -> T(y), forall y: T(y) -> U(y) we can infer forall y: S_2(y) -> U(y) via universal transitivity, which contradicts exists y: ~(S_2(y) -> U(y)). Therefore, the answer is no.