How to prove this theorem in this hilbert system

Question

I want to find a proof for $((\alpha \rightarrow (\beta \rightarrow \gamma))\rightarrow ((\alpha \rightarrow \beta)\rightarrow(\alpha \rightarrow \gamma)))$ with these three axioms:

Ax1: $(\alpha \rightarrow(\beta \rightarrow \alpha))$

Ax2: $(\alpha \rightarrow \beta) \rightarrow ((\alpha \rightarrow (\beta \rightarrow \gamma)) \rightarrow (\alpha \rightarrow \gamma))$

Ax3: $(((\lnot \alpha) \rightarrow (\lnot \beta))\rightarrow(\beta \rightarrow \alpha))$

This axioms make a Hilbert system so for rule of inference we have: $\{A,A\rightarrow B\} \vdash B$ or MP (Modus Ponens) and I already proved DT (Deduction Theorem).

Thanks.

Answer 1

• Assume $(\alpha \to (\beta \to \gamma))$ ... (1)

  • Assume $(\alpha \to \beta)$ ... (2)
  • By Ax2: $(\alpha \to \beta) \to ((\alpha \to (\beta \to \gamma)) \to (\alpha \to \gamma))$ ... (3)
  • By using Modus Ponens on (2), (3): $(\alpha \to (\beta \to \gamma)) \to (\alpha \to \gamma)$ ... (4)
  • By using Modus Ponens on (1), (4): $(\alpha \to \gamma)$ ... (5)

• Thus, by Deduction Theorem on (2), (5): $(\alpha \to \beta) \to (\alpha \to \gamma)$ ... (6)

And finally by one more Deduction Theorem on (1), (6): $(\alpha \to (\beta \to \gamma)) \to ((\alpha \to \beta) \to (\alpha \to \gamma))$