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.

Conjunction elimination

Type	Rule of inference
Field	Propositional calculus
Statement	If the conjunction ${\displaystyle A}$ and ${\displaystyle B}$ is true, then ${\displaystyle A}$ is true, and ${\displaystyle B}$ is true.
Symbolic statement	${\displaystyle {\frac {P\land Q}{\therefore P}},{\frac {P\land Q}{\therefore Q}}}$ ${\displaystyle (P\land Q)\vdash P,(P\land Q)\vdash Q}$ ${\displaystyle (P\land Q)\to P,(P\land Q)\to Q}$

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:

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

and

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

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