The term you're looking for is Propositional Calculus. See the Wikipedia entry for a discussion of "axioms" and rules of inference. There is no one list of accepted axioms, equivalences, and/or rules of inference for natural deduction, as you'll see in the linked entry: that's why the entry refers to "a propositional calculus, or logic...", as there are a varied many of them:
...a propositional calculus or logic (also called sentential calculus or sentential logic) is a formal system in which formulae of a formal language may be interpreted to be representing propositions. A system of inference rules and axioms allows certain formulae to be derived, called theorems; which may be interpreted to be true propositions. The series of formulae which is constructed within such a system is called a derivation and the last formula of the series is a theorem, whose derivation may be interpreted to be a proof of the truth of the proposition represented by the theorem.
You'll find some of the more standard axioms and rules of inference that appear in such a logic discussed later in the entry, along with links to specific systems.
See also the Internet Encyclopedia of Philosophy's entry on Propositional Logic.
Table of Contents:
The Language of Propositional Logic
Syntax and Formation Rules of PL
Truth Functions and Truth Tables
Definability of the Operators and the Languages PL’ and PL”
Tautologies, Logical Equivalence and Validity
Deduction: Rules of Inference and Replacement
Rules of Inference
Rules of Replacement
Conditional and Indirect Proofs
Axiomatic Systems and the Propositional Calculus
Meta-Theoretic Results for the Propositional Calculus
Other Forms of Propositional Logic
References and Further Reading