Consider the following premises. forall x_11: P(x_11) -> Q_1(x_11). forall x_11: S(x_11) -> P(x_11). forall x_11: Q_1(x_11) -> S(x_11). exists x_11: P(x_11) -> S(x_11). Can we infer exists x_11: S(x_11) <-> P(x_11) from them? If possible, name the inference rules being used at each step.
We can infer exists x_11: S(x_11) <-> P(x_11) via the existential biconditional introduction rule. Therefore, the answer is yes.