Modus ponens rule

Question

Prove for the calculus of propositions: $$ A\rightarrow B\mapsto(B\rightarrow C)\rightarrow(A\rightarrow C) $$ I had used axioms and most suitable was this one: $$ X\rightarrow Y\rightarrow(X\rightarrow(Y\rightarrow Z)\rightarrow(X\rightarrow Z)) $$ (axiom2). Then I used Modus ponens rule and got this result: $$ A\rightarrow B,\ A\rightarrow B\rightarrow (A\rightarrow(B\rightarrow C)\rightarrow(A\rightarrow C))\\ A\rightarrow(B\rightarrow C)\rightarrow(A\rightarrow C) $$

Then I didn't understand, how to get $(B\rightarrow C)\rightarrow(A\rightarrow C)$. What axiom should I use or rule?

Answer 1

Using the Deduction Theorem:

1. $A \to B$ --- premise

2. $B \to C$ --- assumed [a]

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

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

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

6. $((A \to (B \to C)) \to (A \to C))$ --- from 1) and 5) by Modus Ponens

7. $A \to C$ --- from 4) and 6) by Modus Ponens.

8. $(B \to C) \to (A \to C)$ --- from 2) and 7) by DT, discharging assumption [a]