I have a reference that says the downward Löwenheim-Skolem theorem does not occur
in intuitionistic logic. In the words of the abstract "even a very powerful
version of intuitionistic set theory does not yield any of the usual forms of a countable
downward Löwenheim-Skolem theorem." 

  Skolem's paradox and constructivism.
	Journal of Philosophical Logic.
	Springer Netherlands.
Issue	Volume 16, Number 2 / May, 1987

https://doi.org/10.1007/BF00257838

Also
page 341 of
A Companion to Metaphysics
 By Jaegwon Kim

 "...there is no intuitionistically acceptable analogue
of the classical downward Löwenheim-Skolem theorem"