Write down a proof for $\bot\Rightarrow q$ in proposition calculus I am given the hint in the question that I will need to use the axiom $(((s\Rightarrow \bot)\Rightarrow \bot)\Rightarrow s)$.
The axioms I am using are $$(s\Rightarrow (t \Rightarrow s)) \\((s\Rightarrow(t\Rightarrow u))\Rightarrow((s\Rightarrow t)\Rightarrow(s\Rightarrow u)) \\ (((s\Rightarrow \bot)\Rightarrow \bot)\Rightarrow s)$$
In a proof every step is either an axiom or deduced by modus ponens.

Using your first two axioms you can prove the deduction theorem. So to prove $\vdash \bot \Rightarrow q$, it's enough to prove $\bot \vdash q$. The hint suggests using the third axiom. With that, you can show that $((q \Rightarrow \bot)\Rightarrow \bot)\vdash q$. So you're done if you can prove $\bot \vdash ((q \Rightarrow \bot)\Rightarrow \bot)$. Can you see how to do this?
