Is there a well known proof that shows solutions of $y^2=3x^4+1$ are only (1,2), (2,7) over positive integers?

I found a theorem from a book 'Diophantine equations', L. J. Mordell, which says

The equation $y^2 = Dx^4+1$ where $D>0$ and is not a perfect square, has at most two solutions in positive integers.

But I can't find any proof in this book, and I tried to find its proof but I failed. Is there anybody knows its proof?

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