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. 

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

