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The background of this question is the talk given by Kevin Buzzard.

I could not find the slides of that talk. The slides of another talk given by Kevin Buzzard along the same theme are available here.

One of the points in the talk is that, people accept some results but whose proofs are not publicly available. (He says this leads to wrong conclusions, but, I am not interested in wrong conclusions as of now. All I am interested is are results which are accepted as true but without a detailed proof, or with only a partial proof.)

What are results that are widely accepted to be true with no detailed proof, or only a partial proof?

I am looking for situations where $A$ has asserted in print that he/she has a proof of $X$, but hasn't published a proof of $X$, and then $B$ publishes a proof of $Y$, where the proof depends on the validity of $X$. For example as in page 20,21,22 of the slides mentioned above.

Edit: Please give reference for the following:

Where the result is announced? Where the result is used?

Edit (made after Per Alexandersson's answer) : I am not looking for "readily available but not formally published". As mentioned by Timothy Chow, "there are many more examples if "readily available but not formally published" counts.".