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 {\displaystyle P} implies a statement Q {\displaystyle Q} and a statement R {\displaystyle R} also implies Q {\displaystyle Q} , then if either P {\displaystyle P} or R {\displaystyle R} is true, then Q {\displaystyle 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 → Q {\displaystyle P\to Q} ", and " R → Q {\displaystyle R\to Q} " and " P ∨ R {\displaystyle P\lor R} " appear on lines of a proof, " Q {\displaystyle Q} " can be placed on a subsequent line.