Negation normal form

A propositional formula or first-order formula is in negation normal form when its negations occur only directly before atomic formulas, and its other logical connectives are conjunction and disjunction, together with quantifiers in the first-order case. Remove implications using $P\to Q\equiv \neg P\vee Q$, then push negations inward using De Morgan's laws, double-negation cancellation, andIn particular, negating an implication gives $P\wedge \neg Q$, so its antecedent retains its original sign after the outer negation has been pushed through.