The statement

$(\mathrm{p} \wedge(\mathrm{p} \rightarrow \mathrm{q}) \wedge(\mathrm{q} \rightarrow \mathrm{r})) \rightarrow \mathrm{r}$

$\equiv(\mathrm{p} \wedge(\sim \mathrm{p} \vee \mathrm{q}) \vee(\sim \mathrm{q} \vee \mathrm{r})) \rightarrow \mathrm{r}$

$\equiv((\mathrm{p} \wedge \mathrm{q}) \wedge(\sim \mathrm{p} \vee \mathrm{r})) \rightarrow \mathrm{r}$

$\equiv(\mathrm{p} \wedge \mathrm{q} \wedge \mathrm{r}) \rightarrow \mathrm{r}$

$\equiv \sim(\mathrm{p} \wedge \mathrm{q} \wedge \mathrm{r}) \vee \mathrm{r}$

$\equiv(\sim \mathrm{p}) \vee(\sim \mathrm{q}) \vee(\sim \mathrm{r}) \vee \mathrm{r}$

$\Rightarrow$ tautology