A direct proof of a statement is a logically correct argument establishing the truth of the statement. A proof by contradiction instead assumes that a statement is false and derives a conclusion which is false, thus proving the original statement using the principle of the excluded middle. Two commonly used proofs by contradiction are showing the $\sqrt{2}$ is irrational and that there are infinitely many primes.

If wishing to prove p⇒q by contradiction, we prove ¬(p∧(¬q)), that is, the hypotheses p are still assumed, and the negation ¬q of the conclusion, and then p∧(¬q) is shown to be false. This is equivalent to direct proof by De Morgan’s Laws.