In propositional logic, conjunction elimination (also called and elimination, ∧ elimination, or simplification) 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:
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.
The conjunction elimination sub-rules may be written in sequent notation:
where and are propositions expressed in some formal system.