This article proposes a model to calculate the reliability function, and the mean residual (remaining) life of a piece of equipment, when its degradation state is not directly observable. At each observation moment, an indicator of the underlying unobservable degradation state is observed, and the monitoring information is collected. The observation process is due to a condition monitoring system where the obtained information is not perfect. For that reason, the observation process doesn't directly reveal the exact degradation state. To match an indicator's value to the unobservable degradation state, a stochastic relation between them is given by an observation probability matrix. It is assumed that the equipment's unobservable degradation state transition follows a Markov chain, and we model it using a hidden Markov model. The Bayes' rule is used to determine the probability of being in a certain degradation state at each observation moment. Cox's time-dependent proportional hazards model is considered to model the equipment's failure rate. This paper addresses two main problems: the problem of imperfect observations, and the problem of taking into account the whole history of observations. Two numerical examples are presented.