# 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
implies a statement
and a statement
also implies
, then if either
or
is true, then
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.

Transformation rules |
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Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

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:

where the rule is that whenever instances of " ", and " " and " " appear on lines of a proof, " " can be placed on a subsequent line.

## Formal notation

The *disjunction elimination* rule may be written in sequent notation:

where is a metalogical symbol meaning that is a syntactic consequence of , and and in some logical system;

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

where , , and are propositions expressed in some formal system.

## References

- "Archived copy". Archived from the original on 2015-04-18. Retrieved 2015-04-09.CS1 maint: archived copy as title (link)
- http://www.cs.gsu.edu/~cscskp/Automata/proofs/node6.html