In the philosophical literature on consciousness and the mind-body problem, the conceivability argument against physicalism is usually taken to support a form of dualism between physicality and phenomenality. Usually, the discussion focuses on the qualitative character of experience, which is what the phenomenal feel of a given experience is like. By contrast, the subjective character of experience, or its individuation to a given first-person subject, tends to be set aside. The aim of this paper is to present a new and more robust version of the conceivability argument for dualism that appeals to the subjective character of experience. Drawing on insights by philosophers in the phenomenological tradition, I conceptualise the first-person subjective character of experience as a transcendental condition of possibility for phenomenality that cannot be reduced to third-person facts about the physical world. Given this, the mind-body problem as it pertains to consciousness does not merely concern the inability of the set of physical facts about a brain state to capture the qualitative character of experience, but concerns the existential issue of why this brain state is accompanied by first-person subjectivity at all. This allows us to reconceive the conceivability argument in a way that presents a stronger case for dualism than the traditional version of the argument.