Restricting universal quantifiers with conditions?

Question

I want to express "For every $x\in\mathbb R$ that has (at least) one $p\in\mathbb Z$ and $q\in\mathbb N$, such that $x=\frac pq$, $x\in\mathbb Q$ is true" with logical quantifiers, just like this $$\bigwedge_{x\in\mathbb R}x\in\mathbb Q~~~\text{(condition missing)}~.$$ But where to put the condition "that has one $p\in\mathbb Z$..." ? $$\bigwedge_{x\in\mathbb R}\bigvee_{\matrix{p\in\mathbb Z\\q\in\mathbb N}}x=\frac pq$$ would be wrong, because not every real number can be written as a quotient of integers, so my question: Is there a notation allowing me to restrict universal quantifiers with a condition?

Answer 1

The relative clause translates to a condition. If there exist such and such $p,q$ then $ x$ is rational. So the full formula has the general form "for all $x$ ((there exist $p,q$ such that etc) -> $x$ so and so)"