You are confusing definitions with theorems.
I can define a 'knurb' to be a number divisible by 17. But it doesn't make sense to ask me to prove that as if it were a theorem... indeed, to prove anything about 'knurbs' you would need to first know what a 'knurb' is ... and that's exactly what a definition does.
Same with your post: we are merely defining what an 'equivalence relation' is
As far as your second question goes:
And also why there is "if.. then" condition is used in symmetry and transitive property why not "if and only if" condition can't be use?
I am wondering if maybe you are asking your question because you might have seen 'if and only if' being used in the context of these definitions, where they indeed regularly occur. Indeed, we can say:
$\sim$ is symmetric if and only if for any $a$ and $b$: if $ a \sim b $ then $ b \sim a$
But notice that the use of the 'if and only if' is used to define 'symmetry', and the 'if ... then ...' in the second half is a claim about the $\sim$ relationship. So these are two different things.
Now, it turns out that I can define symmetry as follows:
$\sim$ is symmetric if and only if for any $a$ and $b$: $a \sim b$ if and only if $b \sim a$
But it is, in a way, just happenstance that we can use an 'if and only if' in the second half here. That is, it just so happens to be true (i.e. this is a theorem) that:
for any $a$ and $b$: if $ a \sim b $ then $ b \sim a$ if and only if for any $a$ and $b$: $ a \sim b $ if and only if $ b \sim a$
On the other hand, if you say that :
$\sim$ is transitive if and only if for any $a$, $b$, and $c$: $a \sim b$ and $b \sim c$ if and only if $a \sim c$
then you have really changed the meaning of 'transitive'. For example, if $a,b,c$ are numbers, then we have that if $a < b$ and $b < c$, then $a < c$. That makes sense right? But it is clearly not true that $a < b$ and $b < c$ if and only if $a < c$: if we pick $a = 1$, $b = 3$, and $c = 2$, then we have $a < c$, but we don't have that $a < b$ and $b < c$!
Finally, let's bring this back full circle. Please note that we can define whatever we want and however we want! So, we could have defined 'transitivity' in the above manner. But note that would have meant that $<$ is not 'transitive'. Indeed, many interesting things would not be 'transitive' under that definition of transitive. ... in other words, 'transitive' would not point out a very interesting property. But with the original definition of transitivity we can point out that $<$ is transitive ... and indeed it expresses an interesting property that $<$ (and many other orderings) have.
So, you could say that there is a reason for defining things we way we do ... that reason being that we feel that that definition serves a useful purpose. We don't have a special word for numbers divisible by $17$ because it serves little purpose: unlike being divisible by $2$ (i.e. 'even'), which is a concept that can be fruitfully employed in many mathematical proofs and theorems, being divisible by $17$ is really not all that useful from the standpoint of gaining some interesting conceptual perspective about numbers or making proofs easier or more intuitive. But that is a far cry from saying that some definition is 'right' or wrong', let alone that we can prove that it is 'right' or 'wrong'.
So, the ultimate point remains: if you want to mathematically prove something, you first need to define things. And that's what they do here with the equivalence relationship.
