This article states that
Every cartesian closed Boolean pretopos is in fact a topos.
Is every cartesian closed Boolean category a topos as well?
]]>Added a “warning” for something that tripped me up: the classifying topos of a classical first-order theory is typically not Boolean, even though the classifying pretopos is Boolean. For a topos to be Boolean is much stronger – as Blass and Scedrov showed, it implies -categoricity.
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