Consider the following problem:
Is $\{\neg p_1 \land p_2 \Rightarrow p_0, p_1 \Rightarrow (\neg p_1 \Rightarrow p_2), p_0 \Leftrightarrow \neg p_2\}$ consistent?
I solved it in the following manner:
Observe that the third hypothesis gives $p_0 \Rightarrow \neg p_2$. At the same time, the first gives $\neg p_1 \land p_2 \Rightarrow p_0$. This can not be true, as will be shown below.
Observe that the hypothesis $\neg p_1 \land p_2$ is cancelled, so $Hip(D)$ is a subset of the set we were required to evaluate.
In truth, the contradiction is no different than that following from a set such as $\{\varphi \Rightarrow \psi, \psi \Rightarrow \neg \varphi\}$, which is clearly inconsistent, albeit somewhat less obvious. The derivation that proves this set is contradictory in fact follows the same structure than the one we did before:
Here too the assumption $[\varphi]_1$ is cancelled.
My confusion comes from the fact that we are not simply deriving a contradiction from the hypotheses. More precisely, we are deriving that certain assumptions that are not in the hypotheses set ($[\neg p_1 \land p_2]_1$ in the first case, $[\varphi]_1$ in the second case) imply a contradiction. I am not sure whether this is correct in proving the hypotheses set is inconsistent, precisely due to the dependence on these extraneous assumptions. I was wondering if someone could clear this out for me.
Thanks in advance.