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