"If any problem in NP is not polynomial-time solvable, then no NP-complete problem is polynomial-time solvable."
Assume that the following were true:

A problem $X$ in NP is proven to not to be able to be solved in polynomial time.
A NP-complete problem $Y$ is proven to be solved in polynomial time with algorithm $Z$.

However, since $Y$ is a NP-complete problem we can transform an instance $x$ of $X$ into an equivalent instance $y$ of $Y$ in polynomial time, and then solve it with $Z$ in polynomial time. The sum of two polynomial times is polynomial, thus our assumption is violated.