Consider the following premises. forall y: S(y) or W_1(y). forall y: U(y) <-> W_1(y). exists y: R(y). forall y: U(y) -> W_1(y). forall y: S(y) -> U(y). Can we infer exists y: U(y) and R(y) from them? If possible, name the inference rules being used at each step.
 From forall y: U(y) <-> W_1(y) we can infer forall y: W_1(y) -> U(y) via universal biconditional elimination. From forall y: U(y) -> W_1(y), forall y: W_1(y) -> U(y) we can infer forall y: U(y) <-> W_1(y) via universal biconditional introduction. From forall y: U(y) <-> W_1(y) we can infer forall y: W_1(y) -> U(y) via universal biconditional elimination. From forall y: S(y) -> U(y), forall y: W_1(y) -> U(y), forall y: S(y) or W_1(y) we can infer forall y: U(y) via universal disjunction elimination. Finally, from forall y: U(y), exists y: R(y) we can infer exists y: U(y) and R(y) via existential conjunction. Therefore, the answer is yes.