Correct use of Axiom Schema of Comprehension

Hello all

I'm going through the book "introduction to set theory" by Hrbace. on page 11 Hrbacek introduces another way on defining a set that builds on the 'Axiom Schema or Comprehension' method

$\displaystyle \{ x \in A | P(x) \}$

I'll post what he writes to explain his method... but to summarise he explains that

$\displaystyle \{ x | P(x) \}$ can be used as long as $\displaystyle P(x)$ implies $\displaystyle x \in A $

Now when he first posted this I was unaware as the point of this. I started 'googling' axiom scheme of comprehension and came across these three posts...

Russell's paradox: how is it resolved?

ZFC and Russell's Paradox

[Russell's Paradox] Help with Notation

From here I learnt that the ASoC was developed by ZFC to get around this Russel Paradox. That is $\displaystyle \{ x | P(x) \}$ where $\displaystyle P(x)=x \notin x$

Here is what he writes...

Quote:

We conclude this section with another notational convention.

Let $\displaystyle P(x)$ be a property of $\displaystyle x$ (and, possibly, of other parameters).

If there is a set $\displaystyle A$ such that, for all $\displaystyle x$, $\displaystyle P(x)$ implies $\displaystyle x \in A$. then $\displaystyle \{x \in A | P(X)\}$ exists, and, moreover, does not depend on A. That means that if $\displaystyle A'$ is another set such that for all $\displaystyle x$, P(x) implies $\displaystyle x \in A'$. THen $\displaystyle \{ x \in A' | P(x) \} = \{ x \in A | P(x) \}$ Prove It

I don't follow his line of reasoning on why these sets are the same and why they exist and dont depend on their set A or A'.

Dylan