Consider the following premises. R(f_1) -> p_2. forall y_3: P_2(y_3) or R(y_3). P_2(f_1) -> p_2. Can we infer ~p_2 from them? If possible, name the inference rules being used at each step.
 From forall y_3: P_2(y_3) or R(y_3) we can infer P_2(f_1) or R(f_1) via universal instantiation. Finally, from P_2(f_1) -> p_2, R(f_1) -> p_2, P_2(f_1) or R(f_1) we can infer p_2 via disjunction elimination, which contradicts ~p_2. Therefore, the answer is no.