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Let's say I have a transaction with txid=A which has bip125-replaceable=True. If I bump the fee the original transaction will be removed from the mempool and replaced by the new one, so how do I know that the new transaction is, in fact, the same as the old one just with a higher fee? Does the txid stay the same? If so, than the answer is obvious. But if it changes, e.g. txid=B, then this is a completely new transaction. How can I establish the link to the old one?

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The txid will change, because the transaction has a different effect. It will have more/larger inputs or less/smaller outputs, possibly combined with other changes.

The link between the two transactions is that they conflict. This means that they directly or indirectly depend on spending the same predecessor output. As every output can be spent exactly once, both cannot end up in the same chain.

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