Find a proof for the following tautology

Question

I was introduced to Axiomatic Theory in last class and I need to know how to solve this kind of problem in the midterm next week. However, I have no idea how to solve these kind of problems. We had no practice in class and we won't have any, so I'm very desperate.

I need to find a proof for $\alpha \rightarrow (\lnot \alpha \rightarrow \lnot \beta)$ using the usual axioms and Modus Ponens

Any help would be greatly appreciated.

Answer 1

Use importation:

$$ p\to(q\to r) \vdash ((p\wedge q)\to r) $$

So,

$$ \alpha\to(\neg\alpha \to \neg\beta) \vdash ((\alpha\wedge\neg\alpha)\to\neg\beta) \vdash (F\to\neg\beta) \vdash T $$