In propositional logic, disjunction elimination (sometimes named proof by cases, case analysis, or or elimination) is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof. It is the inference that if a statement P implies a statement Q and a statement R also implies Q, then if either P or R is true, then Q has to be true. The reasoning is simple: since at least one of the statements P and R is true, and since either of them would be sufficient to entail Q, Q is certainly true. An example in English: It is the rule can be stated as: where the rule is that whenever instances of "P \to Q", and "R \to Q" and "P \lor R" appear on lines of a proof, "Q" can be placed on a subsequent line.

Formal notation

The disjunction elimination rule may be written in sequent notation: where \vdash is a metalogical symbol meaning that Q is a syntactic consequence of P \to Q, and R \to Q and P \lor R in some logical system; and expressed as a truth-functional tautology or theorem of propositional logic: where P, Q, and R are propositions expressed in some formal system.