Could you help me prove next sentence?

Question

I just started learning logic and can't understand how to prove the following expression?

p ⇒ (q ∧ r ), ¬p ⇒ r , p ∨ ¬p I need derive r.

Answer 1

Suppose$ \lnot r$.

Then by $\lnot p \rightarrow r \equiv \lnot r \rightarrow p $, you get $p$. So you get $(q \land r) $ from $p \rightarrow (q \land r)$ and hence $r$, which is a contradiction to the assumption $\lnot r$.

This proves that there is no valuation to make $r$ false. So you can derive $r$.

Answer 2

This is a case of $\vee$ elimination, $\to$ elimination, and $\wedge$ elimination.

$$\dfrac{p\vee \neg p\quad \dfrac{\dfrac{p\;,\, p\to(q\wedge r)}{q\wedge r}{{\to}\mathsf E}}{r}{\wedge\mathsf E}\quad\dfrac{\neg p\;,\, \neg p\to r}{r}{{\to}\mathsf E}}{r}{\vee\mathsf E}$$