Did my tenant break their lease?

[Zoran Domazet]

I have a tenant on a month to month lease and on June 26 I advised the tenant that I would like them to sign a year lease and rent would be increasing effective 8/1. The tenants texted me stating "I will be giving a 30 days notice" instead of signing the new lease terms. I gave them the option to extend the lease another month since they were having a hard time getting approved for the new property and I sent her an email on July 14th stating the month to month lease will end on 8/31/20 and that the last payment will be dude on 8/1/20 which the tenant replied and agreed to the terms. While I was working on the property last week the tenant came over and advised that they have been approved for the new place and will be out by 7/31 giving me less than a week to do the turnover for the property. The tenants have vacated the property as of 7/30. 

Can I hold the tenant to the lease terms they've agreed on and keep their deposit if they don't make the payment on 8/1? Let me know what you guys think as I am unable to fill the property until 9/1 at this point due to the last minute move out notice.