No. You cannot claim a credit on your Delaware return for tax paid to Philadelphia. DE only allows a credit against other state tax.
Delaware says: "If you are a resident of Delaware (or elect to be taxed as one) and pay income taxes to another State on income earned in the other State which is also included in your Delaware taxable income, the law allows you a tax credit against your Delaware income tax. Do not include city wage taxes or county taxes payable with your other State return."
See page 7 of the 200-01 instruction booklet.
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