Consider the following premises. forall x_4: P_4(x_4) -> Q_4(x_4). ~R_4(c). forall x_4: Q_4(x_4) -> R_4(x_4). forall x_4: R_4(x_4) -> P_4(x_4). Can we infer P_4(c) 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), forall x_4: Q_4(x_4) -> R_4(x_4) we can infer forall x_4: P_4(x_4) -> R_4(x_4) via universal transitivity. From forall x_4: P_4(x_4) -> R_4(x_4), forall x_4: R_4(x_4) -> P_4(x_4) we can infer forall x_4: P_4(x_4) <-> R_4(x_4) via universal biconditional introduction. From forall x_4: P_4(x_4) <-> R_4(x_4) we can infer P_4(c) <-> R_4(c) via universal instantiation. Finally, from P_4(c) <-> R_4(c), ~R_4(c) we can infer ~P_4(c) via biconditional elimination, which contradicts P_4(c). Therefore, the answer is no.