# Conjunction elimination

In propositional logic, **conjunction elimination** (also called **and****elimination**, **∧ elimination**,[1] or **simplification**)[2][3][4] is a valid immediate inference, argument form and rule of inference which makes the inference that, if the conjunction *A and B* is true, then *A* is true, and *B* is true. The rule makes it possible to shorten longer proofs by deriving one of the conjuncts of a conjunction on a line by itself.

Transformation rules |
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Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

An example in English:

- It's raining and it's pouring.
- Therefore it's raining.

The rule consists of two separate sub-rules, which can be expressed in formal language as:

and

The two sub-rules together mean that, whenever an instance of " " appears on a line of a proof, either " " or " " can be placed on a subsequent line by itself. The above example in English is an application of the first sub-rule.

## Formal notation

The *conjunction elimination* sub-rules may be written in sequent notation:

and

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

and expressed as truth-functional tautologies or theorems of propositional logic:

and

where and are propositions expressed in some formal system.

## References

- David A. Duffy (1991).
*Principles of Automated Theorem Proving*. New York: Wiley. Sect.3.1.2.1, p.46 - Copi and Cohen
- Moore and Parker
- Hurley