Consider the following premises. exists x_7: P_4(x_7) -> Q_7(x_7). forall x_7: Q_7(x_7) -> P_1(x_7). forall x_7: P_1(x_7) <-> P_4(x_7). Can we infer exists x_7: P_4(x_7) -> P_1(x_7) from them? If possible, name the inference rules being used at each step.
We can infer exists x_7: P_4(x_7) -> P_1(x_7) via the existential transitivity rule. Therefore, the answer is yes.