In propositional logic, disjunction eliminationhttp://www.cs.gsu.edu/~cscskp/Automata/proofs/node6.html (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:
If I'm inside, I have my wallet on me.
If I'm outside, I have my wallet on me.
It is true that either I'm inside or I'm outside.
Therefore, I have my wallet on me.
It is the rule can be stated as:
\frac{P \to Q, R \to Q, P \lor R}{\therefore Q}
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:
(P \to Q), (R \to Q), (P \lor R) \vdash Q
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:
(((P \to Q) \land (R \to Q)) \land (P \lor R)) \to Q
where P, Q, and R are propositions expressed in some formal system.
See also
Disjunction
Argument in the alternative
Disjunct normal form
References
