$1.$ $\alpha$ $\rightarrow$$(\beta \rightarrow \alpha)$ --- (Ak)
$2.$ $(\alpha \rightarrow (\beta \rightarrow \gamma)) \rightarrow ((\alpha \rightarrow \beta ) \rightarrow (\alpha \rightarrow \gamma))$ --- (AS)
$3.$ $(\lnot \beta \rightarrow \lnot \alpha) \rightarrow ((\lnot \beta \rightarrow \alpha) \rightarrow \beta)$ --- (A$\lnot$)
I'm told that I can use any formula to prove something, but if I know that $\vdash \alpha \rightarrow \alpha $ why I can't use it to prove anything?
For example : To prove $\lnot (\alpha \rightarrow \beta)\vdash \lnot \beta$
$1.$ I can write $\lnot \beta \rightarrow \lnot \beta $, or,known $\lnot \alpha \rightarrow \alpha \vdash \alpha $,I can say $(\lnot \lnot \beta \rightarrow \lnot \beta)\rightarrow \lnot \beta$.
An other example : to prove $\lnot \alpha \rightarrow \alpha \vdash \alpha $
$1.$ I could say $\alpha \rightarrow \alpha$, all the solutions I provided are not correct, but that's what I understood, can you explain what I'm missing?