I'm reading the following notes on reduction from circuit-sat to 3-sat http://www.cs.cmu.edu/~avrim/451f11/lectures/lect1108.pdf
On the third page i'm unsure how they arrived at the following
In particular we just replace each statement of the form
$y_{3}=\mathrm{NAND}(y_{1},y_{2})$ as
$(y_{1} \cup y_{2} \cup y_{3})\cap(y_{1} \cup y_{2}' \cup y_{3})\cap(y_{1}' \cup y_{2} \cup y_{3}) \cap (y_{1}' \cap y_{2}' \cap y_{3}')$
I'm quite new to propositional logic, i've read a little bit about how to convert things to CNF so i thougt we should have $y_{3}=(y_{1}' \cup y_{2}')$. I'd appreciate any help on the reasoning behind how they replace the statement.
One way to go about it is to just write down the truth table for both expressions and verify that they are equal (noting that in the first NAND expression, $y_3$ must be equal to the value of NAND$(y_1,y_2)$ or else the statement is false). There are only 8 possible ways to assign 3 variables, you just need to verify in each case that the values of the two Boolean expressions are equal.