# Using the Fitch system, how do I prove $((p \implies q) \implies p)\implies p$?

Using the Fitch system, how do I prove $((p \implies q) \implies p)\implies p$? I started with the hypotheses $(p \implies q) \implies p$ and $\sim p$. However, from these hypotheses I did not get the desired contradiction to solve the question.

Thanks sb45

• If $p$ is not true, then $(p\Rightarrow q)$ is true, so from the hypotheses, $p$ is true, a contradiction. – Questioner Feb 21 '17 at 18:13
• The law is pierce's law, and it is well known to not be constructively provable. So if you don't have double negation elimination, or law of the excluded middle, or something similar, then you can't prove it. If you do have law of the excluded middle, then you prove it the same way you prove every propositional formula. There are $n=2$ boolean variables so there are $2^n$ cases of them being true false. Prove the formula under all $4$ cases the stitch it up with proof-by-cases (also called "or elimination"). – DanielV Feb 21 '17 at 18:21 