Problem: Prove $p \rightarrow (q \vee r), \neg q, \neg r \vdash \neg p$ using Modus Tollens.
I need to prove the validity of the above sequent by using natural deduction. Initially, I didn't read the entire problem, and went the long way by doing an implication-elimination, two negation-eliminations, an or-elimination, and finally a negation-introduction. Then I read the problem and realized I had to use MT, and now I'm stuck. I don't have the greatest grasp on propositional logic and natural deduction. Am I allowed to do the following?
Since I have $\neg q$ and $\neg r$ as my premises, can I use an and-introduction and do $\neg q \wedge \neg r$, which would allow me to immediately use MT to deduce $\neg p$? If not, could someone point me in the correct direction? Thanks.