Conjunction elimination

{{Infobox mathematical statement

| name = Conjunction elimination

| type = Rule of inference

| field = Propositional calculus

| statement = If the conjunction $A$ and $B$ is true, then $A$ is true, and $B$ is true.

| symbolic statement =

1. $\backslash frac\{P\; \backslash land\; Q\}\{\backslash therefore\; P\},\; \backslash frac\{P\; \backslash land\; Q\}\{\backslash therefore\; Q\}$
2. $(P\; \backslash land\; Q)\; \backslash vdash\; P,\; (P\; \backslash land\; Q)\; \backslash vdash\; Q$
3. $(P\; \backslash land\; Q)\; \backslash to\; P,(P\; \backslash land\; Q)\; \backslash to\; Q$

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{{Transformation rules}}

In propositional logic, conjunction elimination (also called and elimination, ∧ elimination,{{cite book | author=David A. Duffy | title=Principles of Automated Theorem Proving | location=New York | publisher=Wiley | year=1991 }} Sect.3.1.2.1, p.46 or simplification)Copi and Cohen{{cn|reason=Without title, this is hardly a useful reference.|date=February 2014}}Moore and Parker{{cn|date=February 2014}}Hurley{{cn|date=February 2014}} 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:

:$\backslash frac\{P\; \backslash land\; Q\}\{\backslash therefore\; P\}$

and

:$\backslash frac\{P\; \backslash land\; Q\}\{\backslash therefore\; Q\}$

The two sub-rules together mean that, whenever an instance of "$P\; \backslash land\; Q$" 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.