# Given (p∨r),(¬q∨r), use the Fitch system to prove (p → q) → r

I am trying, given (p∨r),(¬q∨r) to use the Fitch System in order to prove (p → q) → r). Any ideas on how I should proceed?

• Your goal is of the form $X\to Y$. So start by assuming $X$ (i.e. $p\to q$). Now consider the premise $p\lor r$ (while keeping in mind that the goal is to get $r$). Perform $\lor$-Elim on this premise. The $r$ case is trivial. In the $p$ case, you can get $q$ (from $p\to q$ and $\to$-Elim). Now consider the unused premise. – Git Gud Oct 31 '15 at 16:22
• Firstly, thank you for your response. You told me perform V-Elim on p∨r. But, we dont know neither p is true nor r is true. – Marc Blume Oct 31 '15 at 16:42
• @MarcBlume $p\vee r$ is the first premise. So either $p$ is true or (maybe also) $r$ is true. If we assume the former is false, then the later is true, if we assume the latter is false, then the former is true. Hopefully you find that both cases entail the same conclusion: $r$. – Graham Kemp Nov 2 '15 at 1:57

01. p∨r      premise
02. ¬q∨r     premise
03. p→q    assumption
04. p    assumption
05. q    MP 03 04
06. ¬q assumption
07. ⊥  !!! 05 06
08. r  explosion
09. ¬q→r →intro 06-08
10. r  assumption
11. r→r  →intro 10-10
12. r    ∨elim 02 09 11
13. p→r    →intro 04-12
14. r    assumption
15. r→r    →intro 14-14
16. r      ∨elim 01 13 15
17. [p→q]→r  →intro 03-16