Consider the following premises. R(f) -> t_1. forall x_2: P(x_2) -> R(x_2). r_4 or P(f). r_4 -> R(f). Can we infer ~t_1 from them? If possible, name the inference rules being used at each step.
 From forall x_2: P(x_2) -> R(x_2) we can infer P(f) -> R(f) via universal instantiation. From r_4 -> R(f), P(f) -> R(f), r_4 or P(f) we can infer R(f) via disjunction elimination. Finally, from R(f) -> t_1, R(f) we can infer t_1 via modus ponens, which contradicts ~t_1. Therefore, the answer is no.