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Hi, Would you please look at these sentences? 1.Dad: Gilbert (the pet cat) needs a checkup.
Son: A checkup? Like when I go to the doctor?
2.
Lately, Lena's forearm ached when she rotated her wrist, like when she drove a corkscrew into a bottle of wine. What is the complete form in the "like + when-clause"? Is it "like something when..." Thank you very much.
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(1) is informal conversational English. "like" expresses similarity between two things. It is possible to insert various phrases between "like" and "when", e.g. "Like the times when ~" or "Like the checkups that I have when ~", but there is no unique "answer" to this. I would say that in practice, at least informally, we feel that "a checkup" can be compared directly to "when I go to the doctor" without the need to pin down a specific implied noun or noun phrase to head the second part.

In (2), "like" is used to introduce an example of the sort of thing that made her arm ache. This "like when" is loose or casual phrasing. (2) can be expressed in more proper or formal English using different phrasing such as "such as" or "for example":

Lena's forearm ached when she rotated her wrist; for example, when she drove a corkscrew into a bottle of wine.