Philadelphia does NOT have their own permit. Reciprocal agreements are between the State of PA and the other state. Philly had no say in the matter whatsoever.
Yes! That is the correct answer. (There's some really ambiguous or flat-out incorrect advice in this thread.)
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NOTE TO OP: Be careful with an out-of-state carry permit that's issued from a state where you are NOT a resident. Certain states do NOT recognize a permit from another state where you are NOT, also, a resident.
Pennsylvania - and, therefore, Philadelphia - does NOT have an identical residency requirement; so, in Pennsylvania, you're good-to-go. Philadelphia is categorized by the Federal Government as a, 'city of the first class'. This means that, regardless of residency, if you do NOT have a permit from either Pennsylvania, itself, or from another state that shares LTCF reciprocity with Pennsylvania, then you are NOT allowed to carry concealed and - especially, NOT to carry openly - inside Philadelphia's borders.
(Anywhere else in Pennsylvania, and with or without an LTCF, a legal gun owner is allowed to open carry, BUT, not inside Philadelphia.)
The usual federal and state exceptions to where you cannot carry continue to apply inside Philadelphia: No post offices or other federal office buildings, no state courthouses, (city hall in Philly) no public schools, and be careful inside national parks. There are signs posted at the entrance doors to several of Philadelphia's landmark historic buildings which forbid concealed carry on the premises.
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(The above information is free internet advice; and it, therefore, should be treated that way!)