Landlord claims 30 day notice not received

When I paid a visit to their office weeks later to pay my last days of rent, the owner told me that she never received my notice. The same woman at the front desk seemed to not recall anything and said she never received my e-mail. (I had proof of delivery to the correct address.) They chocked it up to overlooking it during their recent office move and gave me a ‘move-out’ letter (backdated to to the day I gave notice) outlining their move-out procedure and confirming my last day of occupation. I told them I would resend the e-mail to make sure that they had a copy. I called them later and they said that they did not receive that one either. Through research they determined that my e-mails had gone to their spam folder, and they had not checked it. The following day they called me and said that they had spoke to the owner of the apartment, and that he was demanding 30 days notice, so they would have to charge me for the 3 weeks of rent for the time that they had claimed to not have received my notice.

My question is, if they stated that e-mail written notice was acceptable, and I have proof that I complied with that, can they charge me for their mistake of not following up and notifying the landlord? Does their signed ‘move out’ letter backdated to the original date provide me with evidence that they accepted my noticed and made a mistake?