## Frequent Links

# Conjunction elimination

Transformation rules |
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Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

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.

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:

- <math>\frac{P \land Q}{\therefore P}</math>

and

- <math>\frac{P \land Q}{\therefore Q}</math>

The two sub-rules together mean that, whenever an instance of "<math>P \land Q</math>" appears on a line of a proof, either "<math>P</math>" or "<math>Q</math>" 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:

- <math>(P \land Q) \vdash P</math>

and

- <math>(P \land Q) \vdash Q</math>

where <math>\vdash</math> is a metalogical symbol meaning that <math>P</math> is a syntactic consequence of <math>P \land Q</math> and <math>Q</math> is also a syntactic consequence of <math>P \land Q</math> in logical system;

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

- <math>(P \land Q) \to P</math>

and

- <math>(P \land Q) \to Q</math>

where <math>P</math> and <math>Q</math> are propositions expressed in some formal system.