Is “implies” the best symbol when rewriting equations? In my mathematical homework, I usually indicate algebraic rewrites of equations using implication, and the symbol "$\implies$" (LaTeX \implies).  For instance, I might write $$ 3 x - y = 0 \implies 3 x = y \implies x = \frac{y}{3} $$ to mean that, since $3 x - y = 0$, the equivalent equation $3 x = y$ is also true, which then indicates that $x = y/3$ is true.  Is logical implication the correct facility to express rewriting an equation into an equivalent form?  If not, what other concept and symbol would be correct here?

The traditional symbol for "therefore" is $\therefore$ (\therefore in $\LaTeX$). I recommend it in your example, because "$A$ implies $B$" and "$A$ therefore $B$" don't mean the same thing: the former means that if $A$ holds, then so does $B$, the latter means that $A$ holds, and as a consquence of that so does $B$. In your example, with $A \equiv 3x - y = 0$, $B \equiv 3x = y$ and so on, it is the latter reading that you want. For a simpler example: "$0 = 1 \implies 1 = 1$" is true, but "$0 = 1 \therefore 1 = 1$" is false.
(Technically $\therefore$ is just logical conjunction, but with an implied hint that the right conjunct is easily derivable from the left conjunct.)
