Consider the following premises. u <-> S_4(a_19). forall y: S_4(y) -> P_1(y). p -> u. ~P_1(a_19). Can we infer p from them? If possible, name the inference rules being used at each step.
 From forall y: S_4(y) -> P_1(y) we can infer S_4(a_19) -> P_1(a_19) via universal instantiation. From S_4(a_19) -> P_1(a_19), ~P_1(a_19) we can infer ~S_4(a_19) via modus tollens. From u <-> S_4(a_19), ~S_4(a_19) we can infer ~u via biconditional elimination. Finally, from p -> u, ~u we can infer ~p via modus tollens, which contradicts p. Therefore, the answer is no.