Hello,

the task is to prove following theorem in propositional logic's axiomatic system (http://ww2.cs.mu.oz.au/255/lec/subject-prop_axiom.pdf):

$\displaystyle \vdash_{ax} A \rightarrow ((A \rightarrow B) \rightarrow B) $

Some basic assumptions:

For propositions A, B and C following are axioms:

$\displaystyle A \rightarrow (B \rightarrow A)$

$\displaystyle (A \rightarrow (B \rightarrow C)) \rightarrow ((A \rightarrow B) \rightarrow (A \rightarrow C))$

$\displaystyle (\neg A \rightarrow \neg B) \rightarrow (B \rightarrow A)$

The only inference rule is Modus Ponens:

$\displaystyle \vdash_{ax} A $

$\displaystyle \vdash_{ax} A \rightarrow B$

$\displaystyle \Rightarrow \ \vdash_{ax} B$

I tried to use the axioms in many ways, but somehow I couldn't get the right combination. At first, the problem seems quite mechanical, but atleast I didn't know how to achieve the goal. Any help is appreciated. Thanks very much!