I found this, just as I found that.
I found this, just as I did that. [omitted "find"]
equal? Additionally I would like to replace "just as" with "like", but I'm not sure if that's possible.
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They are semantically equivalent, and the second sentence works because do acts as a pro-verb (note: not a proverb) and equates to another verb in the agnate construction, namely found.
You may be able to insert like into that sentences, but only if you would be substituting it for as. That is:
I found this, like I did that.
Note that this might be considered informal, and you can leave the just in:
I found this, just like I did that.
The difference in meaning is that there is a particular emphasis on the second clause where just is used, though the comparison remains in either case.