  We present a rigorous security analysis of Continuous-Variable
Measurement-Device Independent Quantum Key Distribution (CV MDI QKD) in a
finite size scenario. The security proof is obtained in two steps: by first
assessing the security against collective Gaussian attacks, and then extending
to the most general class of coherent attacks via the Gaussian de Finetti
reduction. Our result combines recent state-of-the-art security proofs for CV
QKD with new findings about min-entropy calculus and parameter estimation. In
doing so, we improve the finite-size estimate of the secret key rate. Our
conclusions confirm that CV MDI protocols allow for high rates on the
metropolitan scale, and may achieve a nonzero secret key rate against the most
general class of coherent attacks after 10^7-10^9 quantum signal transmissions,
depending on loss and noise, and on the required level of security.
