Consider the following premises. forall y: R_4(y) -> U(y). forall y: U(y) -> S_2(y). forall y: U(y) -> S_2(y). forall y: S_2(y) -> R_4(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: S_2(y) -> R_4(y), forall y: R_4(y) -> U(y) we can infer forall y: S_2(y) -> U(y) via universal transitivity. From forall y: U(y) -> S_2(y), forall y: S_2(y) -> U(y) we can infer forall y: U(y) <-> S_2(y) via universal biconditional introduction. From forall y: U(y) <-> S_2(y) we can infer forall y: S_2(y) -> U(y) via universal biconditional elimination. From forall y: S_2(y) -> U(y), forall y: U(y) -> S_2(y) we can infer forall y: S_2(y) <-> U(y) via universal biconditional introduction. Finally, from forall y: S_2(y) <-> U(y) we can infer forall y: S_2(y) -> U(y) via universal biconditional elimination, which contradicts exists y: ~(S_2(y) -> U(y)). Therefore, the answer is no.