Consider the following premises. forall x: R_3(x) or ~Q_4(x). forall x: R_3(x) -> Q_4(x). exists x: ~Q_4(x). Can we infer exists x: ~R_3(x) from them? If possible, name the inference rules being used at each step.
 From forall x: R_3(x) or ~Q_4(x) we can infer forall x: Q_4(x) -> R_3(x) via universal rewriting. From forall x: R_3(x) -> Q_4(x), forall x: Q_4(x) -> R_3(x) we can infer forall x: R_3(x) <-> Q_4(x) via universal biconditional introduction. Finally, from forall x: R_3(x) <-> Q_4(x), exists x: ~Q_4(x) we can infer exists x: ~R_3(x) via existential biconditional elimination. Therefore, the answer is yes.