Consider the following premises. R_4(a_10). forall x_10: R_4(x_10) -> S_10(x_10). S_10(a_10) -> r_1. Can we infer ~r_1 from them? If possible, name the inference rules being used at each step.
 From forall x_10: R_4(x_10) -> S_10(x_10) we can infer R_4(a_10) -> S_10(a_10) via universal instantiation. From R_4(a_10) -> S_10(a_10), S_10(a_10) -> r_1 we can infer R_4(a_10) -> r_1 via transitivity. Finally, from R_4(a_10) -> r_1, R_4(a_10) we can infer r_1 via modus ponens, which contradicts ~r_1. Therefore, the answer is no.