There is a prime number $p$ such that $P(p, q)$ is true for all prime numbers $q = p$.
Notice that the given statement goes "there is a prime number $p$" followed by "$P(p, q)$ is true" and then followed by "for all prime numbers $q = p$". However, when writing it in math symbols, I put "$\forall q$" before "$P(p, q)$ is true", correct?
Given the information above, is this negation correct?
Let $\mathbb{P}$ be the set of prime numbers.
$$\neg(\exists p {\in}\mathbb{P} \; \space \forall q {\in} \mathbb{P}\; \space q = p \space \text{ such that } P(p, q))$$
$$\forall p {\in} \mathbb{P}\; \space \exists q {\in} \mathbb{P}\; \space q = p \space \text{ such that } \neg P(p, q)$$
For all prime numbers $p$, there is a prime number $q$ equal to p, for which $P(p, q)$ is false.