What do the prime ideals of $A \otimes_C B$ look like?

Question

Let $A, B, C$ be rings and suppose I have $\phi: C \to A$ and $\psi: C \to B$ so that the tensor product $A \otimes_C B$ makes sense. I want to prove that given prime ideals $P$ of $A$ and $Q$ of $B$ such that $\phi^{-1} (P) = \psi^{-1} (Q)$, there exists a prime ideal $R$ of $A \otimes_C B$ such that $P = i^{-1}(R)$ and $Q = j^{-1} (R)$ where $i: A \to A \otimes_C B$ and $j: B \to A \otimes_C B$. How can I prove that such prime ideal exists? (I am asking this because I want to prove that if I take a fibre product of schemes then I can find a point $z \in X \times Y$ above $x \in X$ and $y \in Y$.)