NAND is not Associative

Theorem
Let $\uparrow$ signify the NAND operation.

Then there exist propositions $p, q, r$ such that:
 * $p \uparrow \paren {q \uparrow r} \not \vdash \paren {p \uparrow q} \uparrow r$

That is, NAND is not associative.

Also see

 * NOR is not Associative