Syntactic proof

Question

I want to prove (false => p) syntactically, every attempt so far has been in vain despite using the deduction theorem or (p=>p). Any hints first would be welcomed.

Answer 1

For clarity, I'll use $\bot$ in place of $\text f$ (i.e. false).

Ax.1) $p \to (q \to p)$

Ax.2) $[p \to (q \to r)] \to [(p \to q) \to (p \to r)]$

Ax.3) $((p \to \bot) \to \bot) \to p$

Proof

1) $[((p \to \bot) \to \bot) \to p] \to [\bot \to (((p \to \bot) \to \bot) \to p)]$ --- from Ax.1 with $\bot$ in place of $q$ and $(((p \to \bot) \to \bot) \to p)$ in place of $p$

2) $\bot \to (((p \to \bot) \to \bot) \to p)$ --- from 1) and Ax.3 by Modus Ponens

3) $[\bot \to (((p \to \bot) \to \bot) \to p)] \to [(\bot \to ((p \to \bot) \to \bot)) \to (\bot \to p)]$ --- from Ax.2

4) $(\bot \to ((p \to \bot) \to \bot)) \to (\bot \to p)$ --- from 2) and 3) by MP

5) $\bot \to ((p \to \bot) \to \bot)$ --- from Ax.1 with $\bot$ in place of $p$ and $(p \to \bot)$ in place of $q$

6) $\bot \to p$ --- from 4) and 5) by MP