Make a truth table that includes all three statements in the argument:
| \(P\) | \(Q\) | \(R\) | \(P \imp Q\) | \(P \imp R\) | \(P \imp (Q \wedge R)\) |
| T | T | T | T | T | T |
| T | T | F | T | F | F |
| T | F | T | F | T | F |
| T | F | F | F | F | F |
| F | T | T | T | T | T |
| F | T | F | T | T | T |
| F | F | T | T | T | T |
| F | F | F | T | T | T |
Notice that in every row for which both \(P \imp Q\) and \(P \imp R\) is true, so is \(P \imp (Q \wedge R)\text{.}\) Therefore, whenever the premises of the argument are true, so is the conclusion. In other words, the deduction rule is valid.