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.



Use MP to prove :

$p, p \to q \vdash q$

and then apply Deduction Th twice.


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