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:
and
The two sub-rules together mean that, whenever an instance of "
Formal notation
The conjunction elimination sub-rules may be written in sequent notation:
and
where
and expressed as truth-functional tautologies or theorems of propositional logic:
and
where