# Axiomatic proof of $\vdash p \rightarrow ((p\rightarrow q) \rightarrow q)$

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.

• – Bram28 Jan 31 at 16:36

$$p, p \to q \vdash q$$