inequality derived from law of total variance

Question

From the law of total variance we know: $$ Var(\theta) = E(Var(\theta|Y)) + Var(E(\theta|Y)) $$ Thus we can have $$ Var(\theta) \geq E(Var(\theta|Y)). $$ However, how could we prove the following: $$ Var(\theta|Y>1) \geq E(Var(\theta|Y)|Y>1). $$ It seems intuitive but I am not confident about that. Also, I don't think $$ E(X)=E(Y) \quad=>\quad E(X|Z)=E(Y|Z). $$