If cold water would cause harm but the water can be safely heated, the cause of danger has been removed, so water-based purification remains required.
If no safe means exists and the risk is credible, the exemption of tayammum can apply.
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If Water Can Be Heated Safely
Tayammum: Valid Reasons, Conditions, and Method
If cold water would cause harm but the water can be safely heated, the cause of danger has been removed, so water-based purification remains required.
If no safe means exists and the risk is credible, the exemption of tayammum can apply.