Consider the following premises. exists x_4: P(x_4) -> Q_4(x_4). forall x_4: P_4(x_4) <-> Q_4(x_4). Can we infer exists x_4: P(x_4) -> P_4(x_4) from them? If possible, name the inference rules being used at each step.
 From forall x_4: P_4(x_4) <-> Q_4(x_4) we can infer forall x_4: Q_4(x_4) -> P_4(x_4) via universal biconditional elimination. Finally, from exists x_4: P(x_4) -> Q_4(x_4), forall x_4: Q_4(x_4) -> P_4(x_4) we can infer exists x_4: P(x_4) -> P_4(x_4) via existential transitivity. Therefore, the answer is yes.