Implication vs Equivalence in proofs I understand the definition of both the implication and equivalence signs. When I get asked to prove something, I will probably have to use both implication and equivalence logic. My question is if it's wrong to use the implication sign when theres actually an equivalence? I don't think it is, but unless it's very obvious I feel like I might miss some equivalences. I'm just worried that I'll lose points for that on my exams.
Edit: I'm also wondering for exercises and tasks in general, not only limited to proofs.
 A: An equivalence $p\leftrightarrow q$ is... equivalent to the conjunction of the two implications $p\to q$ and $q\to p$:
$$
(p\leftrightarrow q)\leftrightarrow((p\to q)\land(q\to p))
$$
is a tautology, logically valid. $p\to q$ means "$p$ only if $q$", and "$q\to p$ means "$p$ if $q$". This is why equivalence is usually pronounced "if and only if", often abbreviated "iff" in writing.
To prove an equivalence between $p$ and $p$, often the best approach is to prove that each of $p$ and $q$ implies the other — that both implications are true. Only when you have proved both implications can you conclude that $p$ and $q$ are equivalent, as in, have the same truth value. The following alternate characterization shows that $p$ and $q$ are equivalent iff they are either both true or both false:
$$
(p\leftrightarrow q)\leftrightarrow ((p\land q)\lor(\neg p\land \neg q)).
$$
A: It's certainly not illogical to write $\implies$ when $\iff$ holds because they're both true. Sometimes it's preferable to do so, to emphasize the direction that you're interested in. When checking for correctness it's better to put $\iff$ wherever possible, as one common error is the "hidden assumption", which often takes the form of an invalid or unproven $\iff$ that should be $\implies$ 
