Consider the following premises. exists x_15: P_14(x_15) <-> Q_14(x_15). forall x_15: P_15(x_15). forall x_15: Q_14(x_15). forall x_15: P_15(x_15) -> Q_15(x_15). Can we infer exists x_15: P_14(x_15) and Q_15(x_15) from them? If possible, name the inference rules being used at each step.
 From forall x_15: P_15(x_15) -> Q_15(x_15), forall x_15: P_15(x_15) we can infer forall x_15: Q_15(x_15) via universal modus ponens. From exists x_15: P_14(x_15) <-> Q_14(x_15) we can infer exists x_15: Q_14(x_15) -> P_14(x_15) via existential biconditional elimination. From exists x_15: Q_14(x_15) -> P_14(x_15), forall x_15: Q_14(x_15) we can infer exists x_15: P_14(x_15) via existential modus ponens. Finally, from exists x_15: P_14(x_15), forall x_15: Q_15(x_15) we can infer exists x_15: P_14(x_15) and Q_15(x_15) via existential conjunction. Therefore, the answer is yes.