Consider the following premises. forall x_4: ~Q_4(x_4). forall x_4: U_3(x_4) <-> Q_4(x_4). forall x_4: Q_4(x_4) -> U_3(x_4). Can we infer exists x_4: U_3(x_4) from them? If possible, name the inference rules being used at each step.
 From forall x_4: U_3(x_4) <-> Q_4(x_4) we can infer forall x_4: U_3(x_4) -> Q_4(x_4) via universal biconditional elimination. From forall x_4: U_3(x_4) -> Q_4(x_4), forall x_4: Q_4(x_4) -> U_3(x_4) we can infer forall x_4: U_3(x_4) <-> Q_4(x_4) via universal biconditional introduction. Finally, from forall x_4: U_3(x_4) <-> Q_4(x_4), forall x_4: ~Q_4(x_4) we can infer forall x_4: ~U_3(x_4) via universal biconditional elimination, which contradicts exists x_4: U_3(x_4). Therefore, the answer is no.