# Disjunction elimination

In propositional logic, disjunction elimination[1][2] (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.

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:

$\frac{P \to Q, R \to Q, P \or R}{\therefore Q}$

where the rule is that whenever instances of "$P \to Q$", and "$R \to Q$" and "$P \or R$" 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:

$(P \to Q), (R \to Q), (P \or R) \vdash Q$

where $\vdash$ is a metalogical symbol meaning that $Q$ is a syntactic consequence of $P \to Q$, and $R \to Q$ and $P \or R$ in some logical system;

and expressed as a truth-functional tautology or theorem of propositional logic:

$(((P \to Q) \and (R \to Q)) \and (P \or R)) \to Q$

where $P$, $Q$, and $R$ are propositions expressed in some formal system.