Prove Something is a Contradiction only by Logical Equivalencies

Question

Having the proposition:
$$
(\lnot p \rightarrow q) \land (\lnot p \rightarrow \lnot q)\land(p \rightarrow q) \land (p \rightarrow \lnot q)
$$
I want to prove this to be contradiction. So far I have that:
$$
p \rightarrow q \equiv \lnot p \lor q
$$
But now I am stuck because I want to proceed with the Distributive Law, but don't know how to apply it in my new situation:
$$
(p \lor q) \land (p \lor \lnot q) \land (\lnot p \lor q) \land (\lnot p \lor \lnot q)
$$

Answer 1

By the distributive rule, we have:

$$\begin{align}(p \lor q) \land (p \lor \lnot q) \land (\lnot p \lor q) \land (\lnot p \lor \lnot q)&\equiv [p \lor (q\land \lnot q)] \land [\lnot p \lor (q \land \lnot q)]\\ \\ &\equiv (p \lor \bot) \land (\lnot p \lor \bot)\\ \\ &\equiv (p \land \lnot p)\\ \\ &\equiv \bot\end{align}$$

Note that $\bot$ means "logically false," aka a "contradiction".