When I have an equation like:
$$f(x,y)\tag1$$
And I use a subsitution $y=6+a$ and $x=9-q$ I get the following equation:
$$f(9-q,6+a)\tag2$$
Question: how do I write that mathematically, to go from the first equation to the second?
I think that I should use (according to the given answer):
$$f(x,y)\space\space\space\Longleftrightarrow\space\space\space f(9-q,6+a)\tag3$$
Or is using the arrow wrong? The question is how do I write $(3)$.
The problem with your writing in (1) and (2) is that these expressions are not equations, as you claim. Using an equivalence would make sense in the following context:
Consider the equation $$f(x,y) = 0. \tag1$$ Subsituting $9-q$ for $x$ and $6+a$ for $y$, one gets the following equivalent equation: $$f(9-q,6+a) = 0\tag2$$
If you insist to use an equivalence, which I would not recommend in this case, you could write:
Setting $x = 9-q$ and $y=6+a$, one gets the following equivalence $$f(x)=0 \iff f(9-q,6+a)=0$$
but (3) as you write it does not make much sense. And once again, I would simply avoid any equivalence symbol in your case.
So one should not use arrows to replace sentences, unless one is aware of the precise signification of the arrows. The symbols of the logical connectors $\Rightarrow$, $\Leftrightarrow$, $\vee$ (denoting the logical "or"), $\wedge$ (denoting the logical "and") are used for closed propositions (sentences that are true or false), or predicates (propositions depending on a variable) but NOT to explain something, not to indicate what you are doing. To do that, it is better to use the appropriate words. And a lot of words in mathematics have really precise meaning actually.
The answer from JE Pin seems good for me. And you can replace $\Leftrightarrow$ by $\Rightarrow$, this is actually not a problem (I do not see the problem of Rebellos about that ?). The $\rightarrow$ is used to denote a limit.
Personally, I would do this:
Suppose $f(x)=y$. Let $x\equiv g(s)$. Then $f(g(s))=y$. The use of the triple equals sign "$\equiv$" denotes that $x$ is equal to $g(s)$ as a definition, not just by construction. Here is an example from my own work:
Which can be simplified to $$-mc_{R} f^{\prime }_{tra}( -c_{R} t) =T\Bigl[\frac{2}{c_{L}} f_{inc}( -c_{L} t) -\left(\frac{1}{c_{R}} +\frac{1}{c_{L}}\right) f_{tra}( -c_{R} t)\Bigr]$$ We can make a change of variable $\displaystyle p\equiv -c_{R} t$ and rearrange to obtain:$$mc_{R} f^{\prime }_{tra}( p) -T\left(\frac{1}{c_{R}} +\frac{1}{c_{L}}\right) f_{tra}( p) =-\frac{2T}{c_{L}} f_{inc}\left(\frac{c_{L}}{c_{R}} p\right)$$ This is a first order linear const-coeff ODE for $f_{tra}$, the solution of which, although generally quite complicated (due to $f_{inc}$ not being a function simply of $p$), is usually at least of closed-form.
