# Show that it is possible to prove every proposition ( $\vdash \gamma$) in the following proof system

How can I prove that $\vdash\gamma$ in my proof system that has the following axioms and inference rules:

Ax1: $\alpha \rightarrow (\beta \rightarrow \alpha)$

Ax2: $(\alpha \rightarrow (\beta \rightarrow \gamma)) \rightarrow ((\alpha \rightarrow \beta) \rightarrow (\alpha \rightarrow \gamma))$

Ax3: $(\lnot \beta \rightarrow \lnot \alpha) \rightarrow (\alpha \rightarrow \beta)$

Ax4: $\alpha \rightarrow \beta$

IR: $\frac{\alpha \rightarrow \beta , \alpha}{\beta}$ (MP)

It is very hard for me to see the solution, I hope someone could help me. Thank you in advance.