Consider the following premises. s_1 -> P_8(a_7). Q_8(a_7) or s_1. forall x_8: P_8(x_8) -> Q_8(x_8). forall x_8: Q_8(x_8) -> P_8(x_8). Can we infer P_8(a_7) from them? If possible, name the inference rules being used at each step.
 From forall x_8: P_8(x_8) -> Q_8(x_8), forall x_8: Q_8(x_8) -> P_8(x_8) we can infer forall x_8: P_8(x_8) <-> Q_8(x_8) via universal biconditional introduction. From forall x_8: P_8(x_8) <-> Q_8(x_8) we can infer P_8(a_7) <-> Q_8(a_7) via universal instantiation. From P_8(a_7) <-> Q_8(a_7) we can infer Q_8(a_7) -> P_8(a_7) via biconditional elimination. Finally, from Q_8(a_7) -> P_8(a_7), s_1 -> P_8(a_7), Q_8(a_7) or s_1 we can infer P_8(a_7) via disjunction elimination. Therefore, the answer is yes.