TITLE: Is “implies” the best symbol when rewriting equations?
QUESTION [8 upvotes]: 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?

REPLY [2 votes]: Further to the first two comments and Rob's answer:

When solving equations, to emphasise that your work is displaying a
chain of equivalent equations, i.e., each step is “reversible”,
the correct symbol is $\iff,$ not
$\implies$:
$$P(x)=Q(x)\implies x\in\{2,4,5\}$$ merely means that
$2,4,5$ are candidate solutions, whereas $$P(x)=Q(x)\iff
x\in\{2,4,5\}$$ means that the solution set is actually—not
merely a subset of—$\{2,4,5\}.\\$

Outside of the context of writing proofs and specifying theorems,
where statements like $$\text{“for each real $x,\quad$ <conditions>
$\implies$ <implication>”}$$ are commonplace, usually the connective
that is actually meant is “therefore” rather than
“implies”, in which
case the correct symbol is $\,\therefore\;,$ not $\implies.$
(Published work spell “therefore” out because peppering mathematical
prose with symbols decreases readability.)
For example, \begin{align}&x=3\\\therefore\; &x\in\mathbb R\end{align} and $$x=3\\x\in\mathbb R$$ are asserting the value of $x$ and (explicitly and implicitly, respectively) deducing that $x$ is real, whereas \begin{align}&x=3\\\implies &x\in\mathbb R\end{align} is merely specifying that $x$ being real is the consequence of a particular case: $x$'s value is not being established and $x$ possibly actually equals, say, $2i.$