Conjunction elimination

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:


 * $$\frac{P \land Q}{\therefore P}$$

and


 * $$\frac{P \land Q}{\therefore Q}$$

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


 * $$(P \land Q) \vdash P$$

and
 * $$(P \land Q) \vdash Q$$

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

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


 * $$(P \land Q) \to P$$

and
 * $$(P \land Q) \to Q$$

where $$P$$ and $$Q$$ are propositions expressed in some formal system.