I am trying to prove that $\lnot[(q \vee r) \wedge ((p \vee q) \wedge (\lnot p \vee r))]$ is a contingency using logical equivalence rules. I have tried various steps and keep getting stuck in loops, or making wrong moves and getting tautology or contradiction.
The last steps I tried were dropping the brackets surrounding all the or's (commutativity) and getting $\lnot[q \vee r \vee p \vee q \wedge (\lnot p\vee r)]$ then followed by:
$$ \lnot[q \vee p \vee r \wedge (\lnot p\vee r)] \mbox{ (idempotent)} $$
$$\lnot[q \vee p \vee r \wedge (\lnot p \vee r)] \mbox{ (commutativity again)} $$
$$\lnot(q \vee p \vee r) \mbox{ (absorption)} $$
For one I feel like I'm making some either useless, or incorrect steps here, as well I don't see how $\lnot(q \vee p \vee r)$ would prove it's a contingency. I've tried looking at examples and other questions related to this topic, but I just can't seem to find anything that makes sense given the question. Any help would be greatly appreciated, thank you!