So in order to understand why this must be true, we have to consider what "equals" means in this context.
Assume you were talking to someone who had never heard of division before in his life. How would you explain why you can write $$\frac{1}{5}=\frac{5}{25}?$$ These fractions are equal precisely because $$25\cdot 1 = 5 \cdot 5.$$ In abstract algebra, this is how we define fractions: the equivalency classes of pairs of numbers $(a,b)$ with $b\not= 0$ under the relation $(a,b)\sim(c,d)\Leftrightarrow ad=cb$. (Note: I use the word "numbers" here loosely; see the last paragraph for more info.) We write these pairs $(a,b)$ as $a/b$ or $\frac{a}{b}$ simply because it's convenient.
So, you want to know whether the division property is a special case of the multiplicative property. Let's see if it is.
Assume for the remainder of the answer that $x\not= 0 $. The multiplicative property is that $$\frac{a}{b}=\frac{c}{d} \hspace{10pt}\Leftrightarrow \hspace{10pt}\frac{xa}{b}=\frac{xc}{d}.$$
First, let's check if that's true, based on the definition of fractions above. $$\frac{a}{b}=\frac{c}{d}\hspace{10pt} \Leftrightarrow \hspace{10pt}ad=cb \hspace{10pt}\Leftrightarrow\hspace{10pt} xad=xcb\hspace{10pt} \Leftrightarrow \hspace{10pt}\frac{xa}{b}=\frac{xc}{d}.$$
Alright, I buy that. Now what about the division property?
$$\frac{a}{b}=\frac{c}{d} \hspace{10pt}\Leftrightarrow \hspace{10pt}\frac{a}{xb}=\frac{c}{xd}.$$
How do we check if this is true?
$$\frac{a}{b}=\frac{c}{d}\hspace{10pt} \Leftrightarrow \hspace{10pt}ad=cb \hspace{10pt}\Leftrightarrow\hspace{10pt} axd=cxb\hspace{10pt} \Leftrightarrow \hspace{10pt}\frac{a}{xb}=\frac{c}{xd}.$$
You can see that this is exactly the same thing. The only difference in the proofs are that $xad=xcb$ and $axd=cxb$, so the two properties are equivalent precisely because multiplication is a commutative operation (that is, $ab=ba$ always). In fact, the division property implies the multiplication property too:
$$\frac{xa}{b}=\frac{xc}{d}\hspace{10pt}\Leftrightarrow \hspace{10pt}xad=xcb\hspace{10pt} \Leftrightarrow \hspace{10pt}axd=cxb\hspace{10pt} \Leftrightarrow \hspace{10pt}\frac{a}{xb}=\frac{c}{xd}.$$
Optional Note: All this comes from the concept of a field of fractions, which you can read about on Wikipedia if you're feeling ambitious. In short, any time you have a set of elements that can be added and multiplied, where every $a,b$ satisfy $ab=ba$ and such that no nonzero $a,b$ satisfy $ab=0$, you can define fractions of that set which work exactly as they do with regular numbers.
The word for making fractions like this is "localization," and you can do it with all kinds of stuff other than numbers, like functions, for example.