Difference between $\equiv$ and $=$? What is the difference between $\equiv$ and $=$?
My thought is, that, when $\equiv$ is used $=$ could have been used as well. The resulting expression would not be wrong, but just take on a slightly different meaning. But what exactly is the relation between those symbols?
To give a practical example, consider those:
$$
=:⟺∀:(x∈\iff∈)
$$
$$
=:=∀:(x∈\iff∈)
$$
$$
=:\equiv∀:(x∈\iff∈)
$$
Or those:
$$
5+7=12
$$
$$
5+7\equiv12
$$
$$
5+7=7+5
$$
$$
5+7\equiv7+5
$$
 A: Unfortunately there are a couple of different uses for $\equiv$, and it can be either stronger or weaker than $=$.
One common meaning is "is identically equal to". A typical case where you would see this is $f(x)\equiv g(x)$, and it is stressing that the functions $f$ and $g$ are equal, rather than just their values being equal for a specific $x$. It's basically the same as "for every $x$, $f(x)=g(x)$". We can also use $\equiv$ to distinguish a relation that always holds, such as $cos^2 x\equiv 1-\sin^2 x$, from an equation to be solved for $x$, such as $\cos x=1-\sin x$.
Another common use is in modular, or "clock", arithmetic. Here we say two integers are congruent modulo $m$ if they differ by a multiple of $m$. The connection to clocks is that two times of 4 o'clock are not necessarily the same time, but they have to be a multiple of 12 hours apart. We write $a\equiv b\pmod m$ for "$a$ is congruent to $b$ modulo $m$". Here $16\equiv 4\pmod{12}$, but of course $16\neq 4$.
A: $\equiv$ means "equal for all values of the variables", while $=$ just means "equal" (perhaps for only some values of the variables). For example, compare $$\cos(\alpha+\beta)\equiv\cos\alpha\cos\beta-\sin\alpha\sin\beta$$ with $$ \cos(\alpha+\beta)=\cos(\alpha-\beta).$$The first statement is true because the two sides are equal for all $\alpha$ and $\beta$, while the second can be true for certain values of $\alpha$ and $\beta$ (namely when one of them is a multiple of $\pi$).
A: The $:$ is crucial here, indicating that what's on the left of $:\Longleftrightarrow$, $:=$ or $:\equiv$ is defined as what's on the right. These three two-character symbols all mean the same thing. But if you ask for a comparison of the meanings of the "naked" $\Longleftrightarrow$, $=$, $\equiv$, well, those are all different.
The first two are easy: $\Longleftrightarrow$ means iff, and $=$ means equals. But $\equiv$ can denote identities (making is stronger than $=$) or equivalence relations (which are weaker than $=$, and often denoted $\sim$, though congruence in particular is always represented with $\equiv$.)
A: $\equiv$ is used for $\text{identity}$ mostly, but $=$ is used for equation
