# Boolean Algebra Distributive Laws

Given that $x\cdot(y+z)=(x\cdot y)+(x\cdot z)$ and $x+(y\cdot z)=(x+y)\cdot (x+z)$, what is the name for the opposite of those rules?

Say I'm trying to prove the opposite, and I need to simplify from the form $(x\cdot y)+(x\cdot z)$ to $x\cdot(y+z)$, what would I write for the rule I used to do it?

Thanks

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I believe the distributive law can be cited for either direction; i.e., since it is an equality, what you refer to as the "opposite" is really the "equivalent", i.e.,: $y = x \iff x = y$.
In other words, you can cite the distributive law as your justification (multiplication distributes over addition in $\mathbb{R}$, assuming x, y, and z are real numbers. Ditto for x, y, z in $\mathbb{Z}$, or in $\mathbb{Q}$, etc.)