If I rent out my primary home and pay rent elsewhere, can the rent I paid be used to offset my rental income?

Let me put into numbers, for example, I collect $4000 rent each month, and the property tax, mortgage interest adds up to $2000 per month. I will not consider the depreciation part in this case, as the depreciation you deducted always comes back when you sell it.  And I pay $2500 rent to rent an apartment. Is there a way to take use the $2500 as rental expense? It is theoretically right that in order to rent out my home to get $4k rental income, I have to pay $2500 cost to find myself a place to live.