Consider the following premises. forall y: S(y) -> T_3(y). forall y: U(y) <-> T_3(y). forall y: ~U(y). forall y: S(y) or R_2(y). Can we infer exists y: ~R_2(y) from them? If possible, name the inference rules being used at each step.
 From forall y: U(y) <-> T_3(y), forall y: ~U(y) we can infer forall y: ~T_3(y) via universal biconditional elimination. From forall y: S(y) -> T_3(y), forall y: ~T_3(y) we can infer forall y: ~S(y) via universal modus tollens. Finally, from forall y: S(y) or R_2(y), forall y: ~S(y) we can infer forall y: R_2(y) via universal disjunctive syllogism, which contradicts exists y: ~R_2(y). Therefore, the answer is no.