Consider the following premises. forall y: S_3(y) -> R_2(y). forall y: S_3(y) or ~R_2(y). Can we infer exists y: ~(S_3(y) -> R_2(y)) from them? If possible, name the inference rules being used at each step.
 From forall y: S_3(y) or ~R_2(y) we can infer forall y: R_2(y) -> S_3(y) via universal rewriting. From forall y: S_3(y) -> R_2(y), forall y: R_2(y) -> S_3(y) we can infer forall y: S_3(y) <-> R_2(y) via universal biconditional introduction. Finally, from forall y: S_3(y) <-> R_2(y) we can infer forall y: S_3(y) -> R_2(y) via universal biconditional elimination, which contradicts exists y: ~(S_3(y) -> R_2(y)). Therefore, the answer is no.