Does $=$ stand for $\iff$ in some way? $\iff$ is a logical connective which can only be applied to pairs of propositions. So I understand that saying $8 \iff 5+3$ doesn't make much sense, because $8$ and $5+3$ are not propositions. However, when we get to less trivial statements the border between $=$ and $\iff$ becomes more fuzzy for me. Consider the following example:


Let $T_v$ be a translation of the complex plane by $v$. Let $R_a^{\phi}$ be a rotation of the plane by $\phi$ around $a$.
In my Complex Analysis book, we proved $R_a^{\phi}(z) = T_k \circ R_0^{\phi}(z)$, where $k = a(1-e^{i \phi})$. That is, any rotation of $z$ around $a$ can be written as a rotation around the origin and the a suitable translation.

To me it seems that in this case, equality can definitely be interpreted as $\iff$. I can say the above statement as: (If I rotate $z$ around $a$, then I get the same result as rotating $z$ around the origin and translating) $\wedge$ (If I rotate $z$ around the origin and do a translation, then I get the same result as  rotating $z$ around $a$).
So what exactly is the relationship between $=$ and $\iff$?
 A: I'll elaborate on my comment and make it in to a answer instead:
I would say that the distinction between '$=$' and '$\Leftrightarrow$' that you made yourself, still holds in your example. The symbol '$=$' denotes a relation between objects while '$\Leftrightarrow$' is a logical connective relating statements that are true or false. 
Remember that a function is a subset of a cartesian product of the domain and codomain, so in your example, $R^{\phi}_a$ and $T_k\circ R^{\phi}_0$ both denote sets, and therefore $R^{\phi}_a=T_k\circ R^{\phi}_0$. Writing $A\Rightarrow B$ for two sets $A,B$, would be literally wrong, but could of course be interpreted as $x\in A \Rightarrow x\in B$.
A: Equality can't be interpreted as a biconditional in your example. "The rotation map… is equal to the translation composed with the other rotation map." Under your interpretation using "iff" instead of "=", the expression would read "A rotation map if and only if a rotation followed by a translation", which is just grammatically wrong.
