Consider the following premises. w or S_3(a_5). forall x_7: P_7(x_7) or ~S_3(x_7). w -> P_7(a_5). Can we infer ~P_7(a_5) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P_7(x_7) or ~S_3(x_7) we can infer forall x_7: S_3(x_7) -> P_7(x_7) via universal rewriting. From forall x_7: S_3(x_7) -> P_7(x_7) we can infer S_3(a_5) -> P_7(a_5) via universal instantiation. Finally, from w -> P_7(a_5), S_3(a_5) -> P_7(a_5), w or S_3(a_5) we can infer P_7(a_5) via disjunction elimination, which contradicts ~P_7(a_5). Therefore, the answer is no.