Is there a compact notation for indicating the reasons for an implication? Is there a compact, commonly used notation for indicating the reasons for an implication? For example, suppose I have previously established or been given $P$, and can use it to show that $B$ follows from $A$ because of $P$. I'd like to be able to write something like 
$$A \underset{P}\Rightarrow  B,$$
but feel like I'm making up notation that will not be understood. Is something like this common practice, and if so are there LaTeX conventions for generating it?
 A: When people write mathematics on a blackboard, they have many idiosyncratic ways of abbreviating arguments to save space. But when we write mathematics in print, we typically use prose. So there is no common way in LaTeX to do what you ask, because we generally just explain the argument in words. 
A: You can use the super useful mathtools package (for bunch of other stuff too!). Use it with \xRightarrow[below]{above}. There is also extarrows package with similar usage \xlongequal[below]{above}. 
My humble advice is not to exploit this, rather keep the text at max from (2) and (5) or via Fubini Thm. or something similar to draw attention to known results, previous relations, elsewhere in the text.
A: If you want to enphasize that you are using P, together with A, to deduce B
you could write 
P & A ⟹ B
that is: if P and A then B
This is standard logic notation and it draws attention to P.
It might also come useful if you or your readers want to use some proof-assistant or automatic-theorem-prover
