Could somebody give a hint how to prove the following theorem $\text{A}\vdash \left( \text{A}\to \text{B} \right)\to \text{B}$ using a Hilbert style proof? Three axiom schemas
(A1) $\alpha \to \left( \beta \to \alpha \right)$
(A2) $\left( \alpha \to \left( \beta \to \gamma \right) \right)\to \left( \left( \alpha \to \beta \right)\to \left( \alpha \to \gamma \right) \right)$
(A3) $\left( \neg \alpha \to \neg \beta \right)\to \left( \beta \to \alpha \right)$
together with Modus Ponens as a rule if inference. $\alpha \to \beta \text{,}\alpha \vdash \beta $.
Edit:
I’m fully aware that this can be easily proven with the Deduction theorem. In fact, it’s almost the very definition of the Deduction theorem. It can also be proven, for example, with a truth table: if both $\text{A}$ and $\text{B}$ are true then, obviously, $\left( \text{A}\to \text{B} \right)\to \text{B}$ is also true; if $\text{A}$ is true but $\text{B}$ is false then $\text{A}\to \text{B}$ is false and $\left( \text{A}\to \text{B} \right)\to \text{B}$ is true, therefore, $\text{A}\vDash \left( \text{A}\to \text{B} \right)\to \text{B}$, but that’s not the point.
I’m struggling to prove this using a Hilbert style proof: $\text{A}\vdash \left( \text{A}\to \text{B} \right)\to \text{B}$. So far, I’ve tried several instances of A1 such as $\text{B}\to \left( \left( \text{A}\to \text{B} \right)\to \text{B} \right)$, $\text{A}\to \left( \left( \text{A}\to \text{B} \right)\to \text{A} \right)$ etc. but none of them worked. I also tried instances of A2 like $\left( \text{A}\to \left( \text{A}\to \text{B} \right) \right)\to \left( \left( \text{A}\to \text{A} \right)\to \left( \text{A}\to \text{B} \right) \right)$.
I’d be really grateful if somebody could give me, let’s say, the first two elements in the proof sequence $\text{A},...,\left( \text{A}\to \text{B} \right)\to \text{B}$ after the premise $\text{A}$. Thank you!
I will answer your original question.
\begin{align*} &\!\!\!\!\!\!\!\!\!\!\!\!\!1. A \tag*{hypothesis}\\ &\!\!\!\!\!\!\!\!\!\!\!2. A \rightarrow ((A \rightarrow B) \rightarrow A) \tag*{(A1)}\\ &\!\!\!\!\!\!\!\!\!\!\!3. (A \rightarrow B) \rightarrow A \tag*{1,2 MP}\\ &\!\!\!\!\!\!\!\!\!\!\!4. (A \rightarrow B) \rightarrow (A \rightarrow B) \tag*{theorem}\\ &\!\!\!\!\!\!\!\!\!\!\!5. ((A \rightarrow B) \rightarrow (A \rightarrow B)) \rightarrow (((A \rightarrow B) \rightarrow A) \rightarrow ((A \rightarrow B) \rightarrow B))\tag*{(A2)}\\ \end{align*} Can you proceed from here?