A single axiom that is satisfied only by NAND or NOR must be of the form "something equals ," since otherwise constant functions would satisfy the equation. With up to six NANDs and two variables, none of the possible axiom systems of this kind work even up to 3-value operators. But with 6 NANDS and 3 variables, 296 of the possible axiom systems work up to 3-value operators, and 100 work up to 4-value operators (Wolfram 2002, p. 809).
Of the 25 of these that are not trivially equivalent, it then turns out that only the Wolfram axiom
and the axiom
where denotes the NAND operator, are equivalent to the axioms of Boolean algebra (Wolfram 2002, pp. 808-811 and 1174). These candidate axioms were identified by S. Wolfram in 2000, who also proved that there were no smaller candidates.