Consider the following premises. forall y: R_3(y) <-> R(y). exists y: R(y) or Q_3(y). forall y: R_3(y) <-> Q_3(y). Can we infer forall y: ~R_3(y) from them? If possible, name the inference rules being used at each step.
 From forall y: R_3(y) <-> Q_3(y) we can infer forall y: Q_3(y) -> R_3(y) via universal biconditional elimination. From forall y: R_3(y) <-> R(y) we can infer forall y: R(y) -> R_3(y) via universal biconditional elimination. Finally, from forall y: R(y) -> R_3(y), forall y: Q_3(y) -> R_3(y), exists y: R(y) or Q_3(y) we can infer exists y: R_3(y) via existential disjunction elimination, which contradicts forall y: ~R_3(y). Therefore, the answer is no.