In practice, Process capability indices such as lifetime performance index C_L indicate the relationships between the actual process performance and the manufacturing specifications, where L is the lower specification limit and it is known. Progressive first-failure censoring scheme is quite useful in many practical situations where lifetime of a product is quite high and test facilities are scarce but test material is relatively cheap. This study, under the assumption of two-parameter exponential distribution and by applying data transformation constructs a uniformly minimum variance unbiased estimator (UMVUE) of C_L based on a progressive first-failure censored sample. Then the UMVUE of C_L is utilized to develop the new hypothesis testing procedure. Finally, two illustrative examples are given to assess the behavior of this test statistic for testing null hypothesis under given significance level.