Tom Evans and Kate James had appealed to the Supreme Court to overturn a decision by the Court of Appeal upholding Alder Hey Hospital's plan to withdraw the 23-month-old's life support. However the British High Court disagreed with their contention today.
Alfie's parents filed a petition to the Supreme Court after the UK Court of Appeal ruled this Monday against the parents' argument that their son was being unlawfully detained by the hospital.
Bishop John Sherrington said the bishops had a "deep awareness of the pain" for Alfie's parents.
Mr Evans and Ms James' lawyers, the Christian Legal Centre, said they will appeal to the European Court of Human Rights, despite the court ruling.
The Supreme Court acknowledged that this was a "desperately sad case...principally of course for Alfie's parents for they love their little boy dearly and want to do all in their power to keep him alive".
"There is also no reason for further delay".
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But on Friday three supreme court justices dismissed the pair's application, agreeing with Alfie's doctors by saying "there is no hope of him getting better". That is the law in this country. Alfie's parents fought the decision to the Supreme Court and before the European Court of Human Rights.
The parents of Alfie Evans have lost the latest round of their legal fight after failing to persuade Supreme Court justices to consider their case for a second time.
After Alfie's parents filed their petition to the Supreme court this Tuesday, Tom Evans flew to Rome via Athens to meet with Pope Francis. A date has been set by Alder Hey Children's Hospital for taking Evans off life-support. The protests at the hospital lasted for five days, until Tom and Kate asked supporters to disperse. "We would want baby Alfie to have the proper, appropriate care for his condition, ' he said".
Alfie suffers from an unknown neurological degenerative condition reducing him to what the hospital called a "semi-vegetable state". Alfie's cognitive abilities were actually improving after months of treatment, Alfie's parents and their legal representation argued.
The court refused to release details about the approved plans for Alfie's eventual death, citing Alfie's privacy.