I found this online (from a Joel David Hamkins's post):
If ZFC is consistent, then there is a model of ZFC in which every set-theoretic object is definable (in the sense of $\emptyset$-definable). This is true in the minimal transitive model of set theory, by observing that the collection of definable objects in that model is closed under the definable Skolem functions of L, and hence by Condensation collapses back to the same model, showing that in fact every object there was definable.
The post was about sets that are definable from ZFC, in the sense of unique solutions of some formulae of the set theory language with a single variable (without parameters). I think that the right way to interpret this is the following: $$\forall M\models ZFC+V=L,\forall x\in M\exists\varphi\big(M\models\varphi(x)\wedge\exists!v_0\varphi(v_0)\big).$$ If it is correct, I can't understand the argument. Can anyone explicit the steps for me?