Consider the following premises. forall x: Q(x) -> R(x). forall x: ~Q_3(x). forall x: Q_3(x) <-> R(x). exists x: Q_4(x) <-> Q(x). Can we infer forall x: Q_4(x) from them? If possible, name the inference rules being used at each step.
 From forall x: Q_3(x) <-> R(x), forall x: ~Q_3(x) we can infer forall x: ~R(x) via universal biconditional elimination. From forall x: Q(x) -> R(x), forall x: ~R(x) we can infer forall x: ~Q(x) via universal modus tollens. Finally, from exists x: Q_4(x) <-> Q(x), forall x: ~Q(x) we can infer exists x: ~Q_4(x) via existential biconditional elimination, which contradicts forall x: Q_4(x). Therefore, the answer is no.