proving that formulas are theorems using deduction theorem

Question

I'm working in $L_0$ I know I have to use the three axioms and inference rule. However, I'm finding it difficult to prove that $(\neg \alpha \rightarrow (\alpha \rightarrow \beta))$ is a theorem. I find it very fiddly and unintuitive - as if I just have to sort of plug in random formulae and combine them to get what I need.

Any advice would be appreciated.

Answer 1

Hint

1) $\vdash \lnot \alpha \to (\lnot \beta \to \lnot \alpha)$ --- Ax.1

2) $\lnot \alpha$ --- assumed [a]

3) $\lnot \beta \to \lnot \alpha$ --- from 1) and 2) by modus ponens

4) $\vdash (\lnot \beta \to \lnot \alpha) \to (\alpha \to \beta)$ --- Ax.3

5) $\alpha \to \beta$ --- from 3) and 4) by modus ponens

6) $\lnot \alpha \to (\alpha \to \beta)$ --- from 2) and 6) by Deduction Theorem, discharging [a].