I'm trying to solve a question which asks me to prove $\vdash p \rightarrow ((p\rightarrow q) \rightarrow q)$ using the axiomatic proof system with modus ponens as it's only rule, the axioms
PL1: $\phi\rightarrow (\psi \rightarrow \phi)$
PL2: $(\phi \rightarrow (\psi \rightarrow \chi))\rightarrow((\phi\rightarrow\psi)\rightarrow(\phi \rightarrow\chi)) $
PL3: $(\text{~}\psi \rightarrow \text{~}\phi)\rightarrow((\text{~}\psi \rightarrow \phi)\rightarrow\psi)$
using the deduction theorem for propositional logic (if $\Gamma, \phi \vdash \psi$ then $\Gamma \vdash \phi \rightarrow \psi$).
I'm really struggling with this, so I'd appreciate any help you could offer.
Best Answer
Hint
Use MP to prove :
$p, p \to q \vdash q$
and then apply Deduction Th twice.