Give a categorical proof of p -> [q -> q]

Question

I am doing Fitch's Exercises of Symbolic Logic, Chapter 1. This is the first exercise. We have so far axioms such as the distributivity axiom, the axiom of conditioned repetition, the transitivity of implication. I really don't know how to show that this is a theorem.

Answer 1

Hint: If you can prove $r$, you can use $r\to (p\to r)$ to prove $p\to r$;.

Answer 2

thanks! I figured out that the proof looks like this (please note that I prove it like this because this is Chapter 1 and so all tools have not been given yet)

1) q->(q->q) axiom of conditioned repetition 2) q->(q->q)->q axiom of conditioned repetition 3) (q->(q->q))->(q->q) by 2 and distributivity 4) q->q by 1, 3, modus ponies 5) p->(q->q) axiom of conditioned repetition