single and double implication arrow? On a test I wrote an implication arrow "$\implies$" to show that I deduced one statement from the previous one, but I didn't get full score since it was more accurate to use an equivalence arrow "$\iff$".
For example:
$$ 2x = 4 \implies x = 2 $$
but it's also true the other way around:
$$ 2x = 4 \impliedby x = 2$$
so it is more correct to write equivalence arrow: 
$$ 2x = 4 \iff x = 2$$
Given this i would assume that if $Q \implies P$ is true, then $Q \impliedby P$ is false. Is this correct?
I don't want to check whether a statement only implies or is equivalent to another every time I do some operations to it. 
So my second question is then: is there some other more loosely defined implication arrow that allows me to show that implication in one direction is true, without saying that implication the other direction is false? I also came across this picture, but i'm not entirely sure what the difference between those two definitions are.
 A: As your own very example shows: just because the implication goes one way doesn't mean that it doesn't go the other way as well. In your case, it goes from left to right and from right to left, so we can write $P \Leftrightarrow Q$.  But this does not mean that one of $P \Rightarrow Q$ or $Q \Rightarrow P$ is false. In fact, both would be true!
There is no commonly used symbol to say that you only have a one-way implication ... You'd have to explicitly say "$P \Rightarrow Q$ but not $Q \Rightarrow P$" ... Or $P \Rightarrow Q$ and $Q \not \Rightarrow P$
A: 
"Given this i would assume that if Q⟹P is true, then Q⟸P is false. 
  Is this correct?"

No that is not implied. However you may be thinking of Modus Tollens which goes:

P⇒Q, notQ hence notP

For example, if it rains (P) then it is wet outside (Q). Since it is not wet outside (notQ) it did not rain (not P). 
Counterexample to the logic you questioned showing it's not correct: 

Assume it is correct and apply to the rain to get:
If it rains then it is wet, hence it's not true that if were wet outside then it has rained. 

That doesn't make any sense when you say it out loud, and that's not an accident. The logic is actually contradictory unless we add additional premises.

Our assumption that it was correct has been refuted. 

The double arrow should be used whenever you have identified things either formally or by definition as one another. 
For example:

You are a bachelor iff you are unmarried. (true by definition)
The linear transformation between two vector spaces is one to one iff that linear transformation only maps one element to 0. (requires formal proof in both directions)

