What are semantic classes that have a syntactic equivalent? This question is related to Benefits for syntactic and semantic classes.
As mentioned there, $\mathsf{PSPACE} = \mathsf{IP}$, which can be interpreted as the semantic class $\mathsf{IP}$ obtaining a syntactic definition.
What are other non-trivial examples for "syntacticizing" a class?
FWIW, the ostensibly semantic class APP defined in [1] was shown to be syntactic in [2].

[1] Valentine Kabanets, Charles Rackoff, Stephen A. Cook, Efficiently
  approximable real-valued functions, ECCC Report TR00-034, 2000.

[2] Emil Jeřábek, Approximate counting in bounded arithmetic, Journal of Symbolic Logic 72 (2007), no. 3, pp. 959–993. doi, preprint