The L deductive System in Propositional Logic I am trying to proof a few statements in the deductive system L, in propositional logic. The system contains 3 axioms (I, II, III below) and a few proven statements (1,2,3,4). In addition, the only inference rule is the modus ponens. In a book I found, the system differs from mine in the 3rd axiom. In the book, the 3rd axiom is (~b->~a)->((~b->a)->b) while my 3rd axiom is  (~b->~a)->(a->b). I wanted to ask, is it possible, given my axioms, statements the MP rule and the deduction theorem, to prove the 3rd axiom from the book? Thank you.
You will need Law of Clavius. For how to derive it using your System L, see help with some Hilbert style proofs in a propositional logic axiom system. Here's an answer to your question.

Theorem $\vdash(\neg\beta\rightarrow\neg\alpha)\rightarrow((\neg\beta\rightarrow\alpha)\rightarrow\beta)$

Proof:


$(\neg\beta\rightarrow\neg\alpha)$ (premise)
$(\neg\beta\rightarrow\alpha)$ (premise)
$(\neg\beta\rightarrow\neg\alpha)\rightarrow(\alpha\rightarrow\beta)$ (Axiom 3 in your System L)
$(\alpha\rightarrow\beta)$ (m.p. 1,3)
$(\neg\beta\rightarrow\beta)$ (Theorem/Statement 4 in your System L, combining 2,4)
$(\neg\beta\rightarrow\beta)\rightarrow\beta$ (Law of Clavius)
$\beta$ (m.p. 5,6) (conclusion)
Therefore $(\neg\beta\rightarrow\neg\alpha),(\neg\beta\rightarrow\alpha)\vdash\beta$ (1,2,7,)
$(\neg\beta\rightarrow\neg\alpha)\vdash(\neg\beta\rightarrow\alpha)\rightarrow\beta$ (deduction theorem)
$\vdash(\neg\beta\rightarrow\neg\alpha)\rightarrow((\neg\beta\rightarrow\alpha)\rightarrow\beta)$ (deduction theorem)
You would need some sort of rule of inference for you to change deductions from I., II., or III. to a form like p -> (q -> (r -> p).  Let '{' and '}' stand in place of the third parenthesis form types, considered by their size. Let '[' and ']' stand in place of the second form types, also considered by their size.  

That might require a rule of inference to change 

{p -> [q -> (r -> p)]} 

into 

p -> [q -> (r -> p)].  

That last step does not accord with the rules of formation.

That all said, something similar does appear to seem possible: