Is the theorem about minimal sufficient statistics by the factorization theorem is a statement of if and only if?

Let theta be an unknown parameter, if for sufficient statistic s(x) i previously found, and for the likelihood's ratio of L(x,theta)/L(y,theta).

If i show that s(x)=s(y) force the likelihood ratio be independent of theta, will i prove minimal of the sufficient statistic? (the opposite direction is of course: the likelihood ratio will force s(x)=s(y))

proofs will be welcomed

Thanks allot