How do I generalize the product for the totient function to a slightly related function?

Question

I already know that
$\phi(n) = n\prod_{p\text{ divides }n}(1-\frac{1}{p})$.
Suppose I define a closely related function $\phi(n,x;i,q)$ s.t.
$\phi(n,x;i,q) = \sum_{1\leq j\leq x, \gcd(j,n)=1,j\equiv i\pmod q}1$
what is a related product for this function? I just need bounds for the product.