[contravariant]

Logic already has separate symbols for 'A implies B' and 'A proves B', where the first can be considered equivalent to '(not A) or B' for all intents and purposes. Since 'A proves B' seems like the very definition of relevance, the only part you could possibly object to is that 'A proves B' implies 'A implies B', but that's given since the alternative would lead to a contradiction where A proves B, but B is false while A is not.

Also, if I understand you correctly you're trying to 'fix' incompleteness by making A -> B undecidable, which seems like it would achieve the opposite.