Generalized "Duality" of Classical Propositional Logical Operations Duality in propositional logic between conjunction and disjunction, $K$ and $A$ means that for any "identity", such as $KpNp = 0$ (ignoring the detail of how to define this notion in propositional logic), if we replace all instances of $K$ by $A$, all instances of $A$ by $K$, all instances of 1 by 0, and all instances of 0 by 1, the resulting equation will also consists of an "identity", $ApNp = 1$.  Suppose that instead of conjunction "$K$" and disjunction "$A$", we consider any pair of "dual" operations $\{Y, Z\}$ of the 16 logical operations such that they qualify as isomorphic via the negation operation $N$, where $Y$ does not equal $Z$.  By "isomorphic" I mean that the sub-systems $(Y, \{0, 1\})$, $(Z, \{0, 1\})$ are isomorphic in the usual way via the negation operation $N$, for example $K$ and $A$ qualify as "isomorphic" in the sense I've used it here. 
If we have an identity involving operations $A_1, \dots, A_x$, and replace each instance of each operation by its "dual" $A'_1$, ..., $A'_x$, replace each instance of 1 by 0, and each instance of 0 by 1, is the resulting equation also an identity?  If so, how does one prove this?  How does one show that the equation obtained via the "duality" transformation here is also an identity?

The resulting equation is also an identity.  This is because any of the $16$ operations can be put in canonical disjunctive normal form using only $\land$, $\lor$, and $\lnot$.  Then the replacement procedure described in the post becomes  the standard one, and we are dealing with ordinary duality. 
