Consider the following premises. forall x_4: R(x_4) <-> Q_1(x_4). exists x_4: S_3(x_4). forall x_4: R(x_4) -> ~S_3(x_4). Can we infer exists x_4: ~Q_1(x_4) from them? If possible, name the inference rules being used at each step.
 From forall x_4: R(x_4) -> ~S_3(x_4), exists x_4: S_3(x_4) we can infer exists x_4: ~R(x_4) via existential modus tollens. Finally, from forall x_4: R(x_4) <-> Q_1(x_4), exists x_4: ~R(x_4) we can infer exists x_4: ~Q_1(x_4) via existential biconditional elimination. Therefore, the answer is yes.