Prove propositional formula is a tautology
Here is what the formula states:


  If $p$ implies $q$, then $p$ and $r$ imply $q$ and $r$.


To prove this, you would like to argue as follows. Assume $p$ implies $q$, and suppose that both $p$ and $r$ hold. Since $p$ implies $q$, then $q$ holds. Hence both $q$ and $r$ hold.

How exactly this proof is expressed depends on your proof system.