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. 
 A: Instead of starting with $x=x$, start with $a=a$ or $b=b$
But yes, sometimes that's exactly what you need to do. For example, suppose I have that $b=a$ and I want to show that $a=b$.  Then I can do:


*

*$b=a \quad $ Premise

*$b=b \quad $ = Intro

*$a=b \quad$  = Elim 1,2

A: 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.

