# How to show that $[(p \rightarrow q) \rightarrow r] \Rightarrow [p \rightarrow (q \rightarrow r)]$

To show that $[(p \rightarrow q) \rightarrow r] \Rightarrow [p \rightarrow (q \rightarrow r)]$ without using a truth table. That is, using logical laws.

• Hint: Try using a truth table. – anakhro Jul 6 '15 at 13:05
• @anakhronizein. Sorry, can you tell me another way not using a truth table? – PHPIsTheBestLanguage Jul 6 '15 at 13:08
• Why is one of your arrows a $\Rightarrow$ and the others are $\rightarrow$? Do they all mean the same? – Ian Jul 6 '15 at 13:09

With Natural Deduction :

1) $(p→q)→r$ --- premise

2) $q$ --- assumed [a]

3) $p→q$ --- from 2) by $\to$-introduction

4) $r$ --- from 1) and 3) by $\to$-elimination

5) $q \to r$ --- from 2) and 4) by $\to$-introduction, discharfging [a]

6) $p \to (q \to r)$ --- from 5) by $\to$-introduction

$[(p→q)→r] \vdash [p \to (q \to r)]$ --- from 1) and 6).

Hint: $p \Rightarrow q \equiv \neg p \vee q$.