Identity regarding Conditional Expected Value

Question

In one of the proofs regarding charachteristic function of XY where X and Y are normally distributed and iid, there is used an identity which states: "One knows that $\mathrm E(\exp(\mathrm itX))=\exp(-\frac12t^2)$ hence $\mathrm E(\exp(\mathrm itXY)\mid Y)=\exp(-\frac12t^2Y^2)$" My question is: how the latter is concluded from the first and can it be generalised?

Answer 1

$$\mathbb E\exp(itXY\mid Y=y)=\mathbb E\exp\left(itXy\mid Y=y\right)=\mathbb E\exp\left(itXy\right)=\exp\left(-\frac12t^2y^2\right)$$where the second equality rests on independence.

This allows the conclusion that:$$\mathbb E\exp(itXY\mid Y)=\exp\left(-\frac12t^2Y^2\right)$$