Consider the following premises. forall x: R(x) <-> R_3(x). forall x: R_3(x) -> R(x). forall x: ~R_3(x). Can we infer exists x: R(x) from them? If possible, name the inference rules being used at each step.
 From forall x: R(x) <-> R_3(x) we can infer forall x: R(x) -> R_3(x) via universal biconditional elimination. From forall x: R_3(x) -> R(x), forall x: R(x) -> R_3(x) we can infer forall x: R_3(x) <-> R(x) via universal biconditional introduction. Finally, from forall x: R_3(x) <-> R(x), forall x: ~R_3(x) we can infer forall x: ~R(x) via universal biconditional elimination, which contradicts exists x: R(x). Therefore, the answer is no.