Getting the spacing around
\forall is tricky. Look at this example:
$\forall W\forall A$ gives
Of course there should be more space after the
W. More interestingly, the spacing between
W is too tight, whereas between
A it is maybe even a bit too loose. Why is this so?
If you use the standard Computer Modern fonts, then
\forall come from the math symbol font
cmsy, whereas letters like
r come from the math italic font
cmmi. Now the point is: TeX doesn't allow automatic kerning of characters from different fonts. But in order to have good spacing after
\forall in all cases, you'd need specific kerning information depending on the subsequent character, as the above example shows.
The kerning information would have to be provided by the font designer, and it would have to be one font containing all the characters involved. I don't know if there are fonts where this is realized.
So yes, for the Computer Modern fonts (and probably most other math fonts) you'll have to rely on manual kerning with
\, and friends to get things right. Here's an example how you could do it. Note that I use
\colon instead of a
:. Moreover, I define
\+ which adds a space slightly smaller than
\exists\+\delta>0\;\forall x,y\in X\colon\dots$