Material implication (rule of inference)

Summary

In classical propositional logic, material implication is a valid rule of replacement that allows a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not-$P$ or $Q$ and that either form can replace the other in logical proofs. In other words, if $P$ is true, then $Q$ must also be true, while if $Q$ is not true, then $P$ cannot be true either; additionally, when $P$ is not true, $Q$ may be either true or false.