Propositional Logic - How to prove that A implies itself?

Question

I'm trying to form a propositional logic proof chain for the tautology $\delta \implies \delta$, using only the axioms

1. $\alpha \implies (\beta \implies \alpha)$
2. $((\alpha\implies (\beta \implies\gamma))\implies((\alpha\implies\beta)\implies( \alpha\implies\gamma)))$
3. $((\neg\beta\implies\neg\alpha)\implies(\alpha\implies\beta))$

The only rule of inference is modus ponens. Seems like it should be easy, but I can't seem to get it. Help appreciated.

Answer 1

1. $(A\to((A\to A)\to A))\to((A\to(A\to A))\to(A\to A))$ - Axiom 2
2. $A\to ((A\to A)\to A)$ - Axiom 1
3. $(A\to(A\to A))\to(A\to A)$ - MP from 1 and 2
4. $(A\to(A\to A))$ - Axiom 1
5. $A\to A$ - MP from 3 and 4