Can I process the part of the effect after "then" in this card's [Main] effect without meeting the "by" condition?
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No, you can't. If you don't trash 1 [X Antibody] or [Chronicle] trait card from your hand, you can't use the part after "then" in this card's [Main] effect to place this card in the battle area.
Source: P-204 Release of the Sealed Knight!