I have the statement:
Let $A_n$ be an indexed set of numbers defined by $A_n = n\mathbb{Z}_{\geq m}$ for $n,m \in \mathbb{Z}$. Consider the claim C: For all $n$, if $x,y$ is in $A_n$, then $x^{(y+1)}$ is in $A_n$.
I wrote C symbolically as the following.
$$C: \forall n, x,y \in A_n \implies x^{(y+1)} \in A_n.$$
Is this symbolic form correct? I feel like I need some separation between the universal quantifier and the rest of the statement. Thanks!
Are the $x,y$ used here arbitrary ("any $x,y$...") or have they been defined earlier ("these $x,y$...")?
If they are predefined constants, then we have: $$\forall n\; \Big(n\in \Bbb Z\wedge x\in A_n\wedge y\in A_n \;\to\; x^{y+1}\in A_n\Big)$$
If they are arbitrary then you have to bind them to a quantified scope.
That is in full: $$\forall n\; \forall x\;\forall y\; \Big(n\in \Bbb Z\wedge x\in A_n\wedge y\in A_n \;\to\; x^{y+1}\in A_n\Big)$$
Or in abbreviated form:
$$\forall n\in\Bbb Z\; \forall x\in A_n\;\forall y\in A_n\; \bigl( x^{y+1}\in A_n\bigr)$$