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

http://www.springerlink.com/content/t28583t748301t04/

Also
page 341 of
A Companion to Metaphysics
 By Jaegwon Kim

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