Consider the following premises. forall x_13: Q_13(x_13) -> P(x_13). exists x_13: P_13(x_13) or Q_13(x_13). forall x_13: P(x_13) <-> Q_3(x_13). forall x_13: P_13(x_13) -> P(x_13). Can we infer exists x_13: Q_3(x_13) from them? If possible, name the inference rules being used at each step.
 From forall x_13: P_13(x_13) -> P(x_13), forall x_13: Q_13(x_13) -> P(x_13), exists x_13: P_13(x_13) or Q_13(x_13) we can infer exists x_13: P(x_13) via existential disjunction elimination. From forall x_13: P(x_13) <-> Q_3(x_13) we can infer forall x_13: P(x_13) -> Q_3(x_13) via universal biconditional elimination. Finally, from forall x_13: P(x_13) -> Q_3(x_13), exists x_13: P(x_13) we can infer exists x_13: Q_3(x_13) via existential modus ponens. Therefore, the answer is yes.