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Manfred comments on Looking for proof of conditional probability - Less Wrong
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I think you need the common-sense axioms P(B|B) = 1 and P(A|all possibilities) = P(A). Given these, the Venn diagram explanation is pretty straightforward.
A Venn diagram is not a proof.
A series of Venn diagrams, with text explanation, is a perfectly fine proof. The pictures can all be translated into statements about sets. The question is what axioms it starts with about P(A|B).