Consider the following premises. forall x_13: P_13(x_13) -> Q_13(x_13). R_13(j_3) <-> P_13(j_3). forall x_13: Q_13(x_13) -> R_13(x_13). ~R_13(j_3). Can we infer P_13(j_3) from them? If possible, name the inference rules being used at each step.
 From forall x_13: P_13(x_13) -> Q_13(x_13), forall x_13: Q_13(x_13) -> R_13(x_13) we can infer forall x_13: P_13(x_13) -> R_13(x_13) via universal transitivity. From forall x_13: P_13(x_13) -> R_13(x_13) we can infer P_13(j_3) -> R_13(j_3) via universal instantiation. From R_13(j_3) <-> P_13(j_3) we can infer R_13(j_3) -> P_13(j_3) via biconditional elimination. From P_13(j_3) -> R_13(j_3), R_13(j_3) -> P_13(j_3) we can infer P_13(j_3) <-> R_13(j_3) via biconditional introduction. Finally, from P_13(j_3) <-> R_13(j_3), ~R_13(j_3) we can infer ~P_13(j_3) via biconditional elimination, which contradicts P_13(j_3). Therefore, the answer is no.