No, "which" is not used as short hand, and this construction is correct; "which" is here a relative pronoun with a special role; instead of standing for a noun phrase, as is often the case, it stands for a whole situation which has been expressed in what precedes. Such clauses introduced by "which" are called sentential relative clauses.
(CoGEL § 15.57) […] Unlike adnominal relative clauses, which have a noun phrase as antecedent, the sentential relative clause refers back to the predicate or predication of a clause, [1] and [2], or to a whole clause or sentence, [3] and [4], or even to a series of sentences [5]:
- They say he plays truant, which he doesn't. [1]
- He walks for an hour each morning, which would bore me. [2]
Relative clauses such as in [1] are used to affirm (if positive) or deny (if
negative) an assertion or thought ascribed to others.
- Things then improved, which surprises me. [3]
- Colin marded my sister and I married his brother, which makes
Colin and me double in-laws. [4]
In [3] the antecedent matrix clause is a single clause and in [4] it is two
conjoined clauses. But one might equally imagine a storyteller coming to the
end of the story with the words:
- which is how the kangaroo came to have a pouch. [5]
Here which could'refer back to the whole length of the story.