We study a BSDE with random terminal time that appears in the modeling of coun-terparty risk in finance. We proceed by reduction of the original BSDE into a simpler BSDE posed with respect to a smaller filtration and a changed probability measure. We relax the basic immersion conditions of the classical reduced-form modeling approach in credit risk by modeling the default time as an invariant time, i.e. a time such that local martingales with respect to a reduced filtration and a possibly changed probability measure, once stopped right before that time, stay local martingales with respect to the original model filtration and probability measure. Using an Azéma supermartingale characterization of invariant times, we establish the equivalence between the original and the reduced BSDE.