Consider the following premises. exists y: ~R(y). forall y: P_3(y) -> R(y). forall y: R(y) -> P_3(y). forall y: S(y) <-> P_3(y). Can we infer exists y: ~S(y) from them? If possible, name the inference rules being used at each step.
 From forall y: P_3(y) -> R(y), forall y: R(y) -> P_3(y) we can infer forall y: P_3(y) <-> R(y) via universal biconditional introduction. From forall y: P_3(y) <-> R(y), exists y: ~R(y) we can infer exists y: ~P_3(y) via existential biconditional elimination. Finally, from forall y: S(y) <-> P_3(y), exists y: ~P_3(y) we can infer exists y: ~S(y) via existential biconditional elimination. Therefore, the answer is yes.