Independency of random variable and sigma algebras

Question

Let:

• $F=({F_t})_{t>0}$ be a filtration of $\sigma$-algebras
• $C_t$ be an $F_{t-1}$-measurable r.v
• $\tilde{Y}_t$ be an $F_{t}$-measurable r.v.

It is sufficent to assume that $\tilde{Y}_t$ is independent on $F_{t-1}$ to state:

$E(C_t\cdot \tilde{Y}_t|F_{0})=E(C_t |F_{0})\cdot E(\tilde{Y}_t)$

Thanks for the answer.