Boolean Algebra: Operation Linkage Laws
The proposition states that if any of the four statements holds, then the other three also holds. This gives in total 12 different theorems. For the proof, you can use truth tables or the axioms of Boolean algebra.

What the comment is trying to explain is that, say, $xy=x$ isn't universally true (consider $x=0,y=1$). However, if $xy=x$ then also $x+y=y$, and vice versa. This is in contrast to identities such as $x+x=x$, which always hold (for all values of $x$).