My task is to rewrite the claim
The product of any four consecutive natural numbers is equal to some perfect square minus 1
Is the statement I have written accurate? $$\forall n\in\Bbb N,\ \exists x\in\Bbb N,\ ((n)(n+1)(n+2)(n+3)=x)\implies(\exists y\in\Bbb N,\ x=y^2-1)$$
I'd appreciate any advice regarding my answer.
Let's look at this one:
$$\forall n\in\Bbb N,\ \exists x\in\Bbb N,\ ((n)(n+1)(n+2)(n+3)=x)$$
It's actually true: it's sufficient to take $x=n(n+1)(n+2)(n+3)$.
Then, your $$\exists y\in\Bbb N,\ x=y^2-1$$ doesn't give a sense to $x$.
Perhaps what you wanted to write is:
$$\forall x\in\Bbb N(\ \exists n\in\Bbb N,\ ((n)(n+1)(n+2)(n+3)=x)\implies(\exists y\in\Bbb N,\ x=y^2-1))$$
which is also a correct answer, a bit longer than the one proposed by Sebastien Shutz.