$\Sigma$ is a set of sentences, the set $\mathcal{L}$ consists of all axioms of the forms:
A1) $\phi \rightarrow (\psi \rightarrow \phi)$
A2) $(\phi \rightarrow (\psi \rightarrow \theta)) \rightarrow ((\phi \rightarrow \psi) \rightarrow (\phi \rightarrow \theta))$
A3) $((\lnot \phi \rightarrow \psi) \rightarrow (( \lnot \phi \rightarrow \lnot \psi) \rightarrow \phi))$
*I can only use $\rightarrow$ with modus ponens.
I need to prove this and I would like some hints. I can get by Deduction theorem, $\Sigma \vdash \phi \rightarrow \theta$ and $\Sigma \vdash \lnot\phi \rightarrow \theta$. I think you need to use A3 with $\phi$ as $\theta$, but I'm not quite sure what else.
For deduction theorem $\Sigma \vdash \phi \rightarrow \theta$ and $\Sigma \vdash \neg \phi \rightarrow \theta$, then $\Sigma \vdash \phi \wedge \neg \phi \rightarrow \theta$, but $\Sigma \not\vdash \phi \wedge \neg \phi$.