Is this proof style legitimate? Normally for direct proof of equality we have the form:   
Prove
$$a = b$$  
Proof (Style):  
We start with $a$ (or $b$) and show through a sequence of logically connected steps that $a$ is $b$ (or the other way around). $_{_\square}$
But, since I'm not great with proofs I just wanted to have someone validate the following direct proof style, or comment on it's relative legitimacy compared with the first proof style:   
Prove 
$$a = b$$
Proof (Style):
$$x = x$$
$$\vdots \tag{logical steps}$$
$$a = b$$  
Furthermore, could someone recommend an elementary text concerning the validity of proof methods (or whatever it's actually called). Thank. 

The second proof is logically correct too, but since you are talking about proof style I wouldn't call it good or preferable. In this aspect, I would like to cite a piece about writing proofs forward from the famous article How to write Mathematics by Paul Halmos:

A familiar trick of bad teaching is to begin a proof by saying: ``Given $\epsilon$, let $\delta$ be $\left(\frac{\epsilon}{3M^2+2}\right)^{1/2}$''. This is the traditional backward proof-writing of classical analysis. It has the advantage of being easily verifiable by a machine (as opposed to understandable by a human being), and it has the dubious advantage that something at the end comes out to be less than $\epsilon$, instead of less than, say, $\left(\frac{(3M^2+7)\epsilon}{24}\right)^{1/3}$. The way to make the human reader’s task less demanding is obvious: write the proof forward. Start, as the author always starts, by putting something less than $\epsilon$, and then do what needs to be done — multiply by $3M^2 + 7$ at the right time and
  divide by $24$ latter, etc., etc. — till you end up with what you end up with. Neither arrangement is elegant, but the forward one is graspable and rememberable.

