Is tenant's notice required even after landlord notice?

Hi,

I am on assured shorthold tenancy agreement (rolling contract). My landlord gave us a 2-month notice to leave the house in May end; So the notice expires July end. I found a new house on 15th of June and would like to move on 5th of July. I informed my landlord of this on 15th of June. He says that's fine, but I would need to pay the rent till 14th of July, since I was also needed to give a on month notice.

So my question is, if landlord has already given 2-month notice and I want to leave early, do I still have to give one month notice (even though all dates are falling within the landlord's 2 month notice period)?

Thanks for your time. Looking forward to hear views.

Best regards.