Going down for flat extensions

Statement

Suppose $φ : R \to S$ is a flat extension i.e. $R$ is a subring of a commutative unital ring $S$ and $S$ is flat as a $R$ -module. Then, if $P_{1} \supset P_{2}$ are prime ideals of $R$ , and $Q_{1} \in S p e c (S)$ contracts to $P_{1}$ , there exists $Q_{2} \in S p e c (S)$ such that $Q_{1} \supset Q_{2}$ , and $Q_{2}$ contracts to $P_{2}$ .