Using DeMorgan’s theorem and the Rules and laws of Boolean algebra proving this should be an easy task. Figure 1-11 shows the equivalency between the basic logic gates and their NAND logic circuits counterpart. Similarly, Figure 1-12 shows the equivalency between the basic logic gates and their NOR logic circuits counterpart.

Figure 1-11 NAND Equivalent Circuits

Note that i.e the complement of "A NAND B"