I have been trying to answer the question "Show that the third axiom cannot be deduced from the other two.", where the first axiom is $p \Rightarrow(q \Rightarrow p)$ , the second axiom is $(p \Rightarrow (q \Rightarrow r)) \Rightarrow ((p \Rightarrow q) \Rightarrow (p \Rightarrow r))$ and the third axiom is $((p \Rightarrow \bot) \Rightarrow \bot) \Rightarrow p$.
I think I have come up with a proof, but I'm unsure as to it's validity, so please tell me what you think. If it is incorrect, please do not tell me a valid proof as I still would like to figure this out for myself.
Proof:
If the 3rd axiom can be deduced from the first two axioms, then it can be deduced from a system whose only rules of inference are the deduction theorem and modus ponens, as the first 2 axioms can be trivially deduced from these rules. However, $$\{(p \Rightarrow \bot) \Rightarrow \bot\}\vdash p $$ is the only logical statement that is equivalent (by the Deduction Theorem) to $$\vdash((p \Rightarrow \bot) \Rightarrow \bot) \Rightarrow p$$.
As modus ponens requires at least 2 initial propositions to infer new propositions, neither of these situations allow for further deductions, so there is no way to construct a proof from these 2 inference rules, and thus the 3rd axiom cannot be deduced from the first two.