In the book of Szabo "Algebra of Proofs", Definition 13.1.9 introduces an elementary topos as a cartesian closed category with a subobject classifier. On the other hand, many other sources including Johnstone add to this definition that the category should contain limits of finite diagrams. For the proof that the requirement on limits of finite diagrams can be removed, Szabo refers the reader to the paper "Colimits in topoi" by Robert Pare who writes in the second paragraph of the section "Preliminaries on topoi" that the existence of finite limits follows from the existence of equalizers which can be derived from appropriate application of the subject classifier. But for finding a monomorphism from the subobject classifier we should have the corresponding pullbacks in the category. Why such pulbacks exist? The definition of a subobject classifier works only in one direction: given a monomorphism it yields the characteristic morphism. But for the opposite direction (from a characteristic morphism to a monomorphism) the definition does not say anything on the existence of the corresponding pullbacks.
The Question. Is it true that a cartesian closed category with a subobject classifier indeed has pullbacks?
If yes, could you provide a (desirably simple) proof? Thank you.