Substitution rules allow us to replace equivalent substatements inside larger formulas. We now apply those rules to conditional statements, especially implications. The negation of an implication is one of the most important formulas in logic because it explains how to deny an if-then statement correctly. In this lesson, negation and duals of conditional statements will be expressed using equivalence laws and De Morgan's Laws.

For statements $p$ and $q$, $$\begin{equation*} \neg(p\to q)\Leftrightarrow p\land\neg q. \end{equation*}$$ This formula says that the negation of an implication asserts the hypothesis and denies the conclusion.