Consider the following premises. exists x: W(x) -> Q(x). forall x: R(x) -> W(x). forall x: R(x) <-> R_2(x). forall x: Q(x) -> R_2(x). Can we infer exists x: Q(x) -> R(x) from them? If possible, name the inference rules being used at each step.
 From forall x: R(x) <-> R_2(x) we can infer forall x: R_2(x) -> R(x) via universal biconditional elimination. From forall x: R(x) -> W(x), exists x: W(x) -> Q(x) we can infer exists x: R(x) -> Q(x) via existential transitivity. From forall x: Q(x) -> R_2(x), forall x: R_2(x) -> R(x) we can infer forall x: Q(x) -> R(x) via universal transitivity. From forall x: Q(x) -> R(x), exists x: R(x) -> Q(x) we can infer exists x: Q(x) <-> R(x) via existential biconditional introduction. Finally, from exists x: Q(x) <-> R(x) we can infer exists x: Q(x) -> R(x) via existential biconditional elimination. Therefore, the answer is yes.