Consider the following premises. forall x_21: P_14(x_21) -> Q_21(x_21). forall x_21: P_7(x_21) -> P_14(x_21). forall x_21: P_21(x_21) -> R_1(x_21). exists x_21: P_7(x_21). forall x_21: P_21(x_21) <-> Q_21(x_21). Can we infer exists x_21: R_1(x_21) from them? If possible, name the inference rules being used at each step.
 From forall x_21: P_21(x_21) <-> Q_21(x_21) we can infer forall x_21: Q_21(x_21) -> P_21(x_21) via universal biconditional elimination. From forall x_21: P_14(x_21) -> Q_21(x_21), forall x_21: Q_21(x_21) -> P_21(x_21) we can infer forall x_21: P_14(x_21) -> P_21(x_21) via universal transitivity. From forall x_21: P_7(x_21) -> P_14(x_21), exists x_21: P_7(x_21) we can infer exists x_21: P_14(x_21) via existential modus ponens. From forall x_21: P_14(x_21) -> P_21(x_21), forall x_21: P_21(x_21) -> R_1(x_21) we can infer forall x_21: P_14(x_21) -> R_1(x_21) via universal transitivity. Finally, from forall x_21: P_14(x_21) -> R_1(x_21), exists x_21: P_14(x_21) we can infer exists x_21: R_1(x_21) via existential modus ponens. Therefore, the answer is yes.