Going down for flat extensions

Statement
Suppose $$\varphi:R \to S$$ 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 $$Q_1 \in Spec(S)$$ contracts to $$P_1$$, there exists $$Q_2 \in Spec(S)$$ such that $$Q_1 \supset Q_2$$, and $$Q_2$$ contracts to $$P_2$$.