Proving a statement using Hilbert's axioms and rules of inference for propositional logic

Question

I want to prove the following statement:

using Hillbert's axioms sand rules of inference for propositional logic. I am having trouble doing so. It seems just like the transitive statment, can I prove it using MP ?

Edit: These are the Hillbert's axioms : 4

Thanks

Answer 1

(1) $A \to B$ --- premise

(2) $B \to C$ --- premise

(3) $\vdash (B \to C) \to (A \to (B \to C))$ --- Ax.1

(4) $A \to (B \to C)$ --- from (2) and (3) by Modus Ponens

(5) $\vdash (A \to (B \to C)) \to ((A \to B) \to (A \to C))$ --- Ax.2

(6) $(A \to B) \to (A \to C)$ --- from (4) and (5) by Modus Ponens

(7) $A \to C$ --- from (1) and (6) by Modus Ponens.

Thus, with (1), (2) and (6), we have :

$A \to B, B \to C \vdash A \to C$.

With two applications of the Deduction Theorem, we have :

$\vdash (A \to B) \to (( B \to C) \to (A \to C))$.