Proof of transitivity in Hilbert Style
I assume you can use modus ponens as a deductive rule. Here is a Hilbert-style proof. As you can see, there is no reason to use the deduction theorem.
- $A \to B$ [assumption]
- $B \to C$ [assumption]
- $(B \to C) \to (A \to (B \to C))$ [by A1]
- $A \to (B \to C)$ [modus ponens, 2 and 3]
- $(A \to (B \to C)) \to ((A \to B) \to (A \to C))$ [by A2]
- $(A \to B) \to (A \to C)$ [modus ponens, 4 and 5]
- $A \to C$ [modus ponens, 1 and 6]