Consider the following premises. exists y: U_4(y) <-> S(y). forall y: S_3(y) -> U_4(y). forall y: Q_2(y) or S_3(y). forall y: Q_2(y) -> U_4(y). Can we infer exists y: S(y) from them? If possible, name the inference rules being used at each step.
 From exists y: U_4(y) <-> S(y) we can infer exists y: U_4(y) -> S(y) via existential biconditional elimination. From forall y: Q_2(y) -> U_4(y), forall y: S_3(y) -> U_4(y), forall y: Q_2(y) or S_3(y) we can infer forall y: U_4(y) via universal disjunction elimination. Finally, from exists y: U_4(y) -> S(y), forall y: U_4(y) we can infer exists y: S(y) via existential modus ponens. Therefore, the answer is yes.