When I was studying quantifiers and its notation, I found that in general, if given a domain of discourse $D$:
We can express the universal quantification as $$\forall x \in D \space P(x) \iff \forall x (x \in D \rightarrow P(x))$$ And we can express the existential quantification as $$\exists x \in D \space P(x) \iff \exists x (x \in D \space \wedge \space P(x))$$
However, my question is that, what if I wanted to restrict the given domain of discourse?
For instance, if I wanted to express "the square of a negative real number is positive", how would I be able to express $\forall x \in \mathbb{R}$ and $\forall x < 0$ at the same time in one expression?
Here is my attempt:
$$\forall x \in \mathbb{R} \space \mid \space \forall x< 0 \space P(x) \iff \forall x(x \in \mathbb{R} \space \mid \space x < 0 \rightarrow P(x))$$
I would appreciate any help! Thank you!