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Math/logic symbol?

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I''m reading a book about mathematical logic and I can find nowhere mentioned the definition of the vertical bar with two horizontal lines before both A''s in the thrid theorem in this picture: As this is the case I presume that it must have a mroe common definition in the math world? Thanks in advance.
Wow.

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I think Theorem 3 reads: "If proposition A is true and it is true that proposition A implies proposition B, then proposition B is true."

If your book doesn''t explain the notation clearly at the beginning, read a more introductory book first.

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Ah, to end that thought I believe it means that the proposition/equation or part therein is valid. Thanks anyways all.


Wow.

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