Shevuos, Chapter 8, Mishnah 5. This Mishnah resembles the Mishnayos at the beginning of the chapter, in which one who swears falsely is not liable, even though what he said was untrue, because his oath did not exempt him from anything. The oath removed no financial liability from him.
The case in the Mishnah:
The owner asks the one who borrowed the item from him: "Heichan shori" - "Where is my ox, the ox that you borrowed?" and the borrower answers "meis" - it died. Now, if it had indeed died, he would be obligated to pay. But in fact it did not die; rather it was broken, or captured, or stolen, or lost. And so too with all the other variations:
"Meis" - it died, but in fact it was broken, captured, stolen or lost.
"Nishbar" - it was broken, but in fact it died, or was captured, stolen or lost.
"Nishbah" - it was captured, but in fact it died, or was broken, stolen or lost.
"Nignav" - it was stolen, but in fact one of the other possibilities occurred.
"Avad" - it was lost, but in fact it died, or was broken, captured or stolen.
In all of these cases, of course, he is obligated to pay.
The claimant, that is, the owner, demands that he swear, and he answers "amen". It is worth noting that he is not obligated to take such an oath at all, since he admitted his liability: everything he said was something for which he is obligated to pay. Rather, in order to show that his words are true, he is willing and agrees to swear to them, and so he answers "amen".
The reason for the exemption:
He is exempt from the oath offering. Even though he swore falsely, this oath has no effect on the financial side: according to his own words, that is, according to his false declaration, he is obligated, and according to the reality he is obligated as well. It turns out that the oath changed nothing, it did not exempt him and it did not release him from anything, and therefore he is exempt from the oath offering.
In summary: in this Mishnah we learned that one who swears falsely about the circumstances in which the borrowed item was lost - in any of the exchanges among died, broken, captured, stolen and lost - is exempt from the oath offering, because both according to his claim and according to the reality his financial obligation remains in force, and the oath exempted him from nothing.