Tautology (rule of inference)

In propositional logic, tautology is one of two commonly used rules of replacement. The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs. They are:

The principle of idempotency of disjunction:

and the principle of idempotency of conjunction:

Where "${\displaystyle \Leftrightarrow }$" is a metalogical symbol representing "can be replaced in a logical proof with."

Theorems are those logical formulas ${\displaystyle \phi }$ where ${\displaystyle \vdash \phi }$ is the conclusion of a valid proof, while the equivalent semantic consequence ${\displaystyle \models \phi }$ indicates a tautology.

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