# Thread: prove by rules of inference

1. ## prove by rules of inference

I have premises
(1) $\displaystyle p$
(2) $\displaystyle p\rightarrow q$
(3) $\displaystyle r$
and conclusion
($\displaystyle p\vee q$) $\displaystyle \rightarrow$ r

I've checked that the argument is valid.
But I still cannot find a way to show it by using rules of inference.

2. ## Re: prove by rules of inference

Which exactly system of inference rules are you using? See this sticky thread.

4. ## Re: prove by rules of inference

Originally Posted by deniselim17
I have premises
(1) $\displaystyle p$
(2) $\displaystyle p\rightarrow q$
(3) $\displaystyle r$
and conclusion
($\displaystyle p\vee q$) $\displaystyle \rightarrow$
From (3) we get $\displaystyle \neg \left( {p \vee q} \right) \vee r$ by addition.

Then by material implication we get $\displaystyle \left( {p \vee q} \right) \to r$.

5. ## Re: prove by rules of inference

Another way is to have $\displaystyle p\lor q\vdash r$ just because r is a premise and use the Deduction theorem (or Conditional Introduction).

6. ## Re: prove by rules of inference

So, premise (1) and premise (2) are not really necessary here?

7. ## Re: prove by rules of inference

That's right.