Proof in axiomatic system (Prop. logic)
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!