Why does 3-way authentication need the 3rd nonce to be stored?

In the following 3-way authentication protocol:

A -> B: $$N_a$$ || B || Signature_KprivA

B -> A: $$N_a+1$$ || $$N_b$$ || A || Signature_KprivB

A -> B: $$N_b+1$$ || Signature_KprivA

($$N_a$$ and $$N_b$$ are nonces, A and B are Alice and Bob, respectively)

How does the presence of the 3rd message remove the requirement to store nonces? In 3-way authentication, if there is no 3rd message, then the nonces must be stored and must have time stamps and the clocks must be synchronized.

I understand that the 3rd message is required for B to be sure that she is communicating with A and not with a MitM. However, in the presentation slides of my class, it's stated that without the 3rd message nonces must be stored-- but with it the nonces don´t need to be stored. Why is that?