Should he mortgage his house to a non-Islamic bank so that he can buy another house?

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Check below answers in case you are looking for other related questions:
Using interest to pay for tuition
Depositing money in riba-based banks out of necessity, in order to protect the money

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Related Answers:
He has a shop in the market and people ask him where riba-based banks are; should he tell them?
He gave his car to someone to sell it and buy a newer model for him. Is this riba?
Where should he deposit his money in these times when riba is so widespread?
Using interest to pay for tuition
Depositing money in riba-based banks out of necessity, in order to protect the money

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