I think you are asking if a route in the routing table, once redistributed, will have the same AD in the local routing table. Is this right? In your example, you have an EIGRP external, with an AD of 170, redistributed into BGP, and your question is, which AD will it take.
First, I'll start by saying BGP is generally an exception to any such rule. :-) In general, however, the route receiving the redistributed route from the routing table does not reinstall the route in the routing table--it would just advertise the route to its peers. Think of it this way:
EIGRP puts a route in the routing table (RIB). BGP redistributes from the routing table, and advertises the route to the BGP peer. BGP wouldn't reinstall the route in the routing table. Suppose it did, what would happen? Suppose that instead of EIGRP and BGP you were dealing with EIGRP and OSPF. OSPF picks up an external EIGRP route, with an admin distance of 170. It puts it in its local table as an OSPF route, and installs it in the routing table, with and admin distance of 110.
The EIGRP route has now been pulled from the table, so the source of the redistributed OSPF route is gone. So, OSPF removes the route, and EIGRP reinstalls it. Now, OSPF sees the route again, picks it up, and installs it. Wash, rinse, repeat, causing havoc in the router and your network. So, generally, speaking, a routing protocol that learns a route from the routing table through redistribution will not attempt to reinstall that route in the routing table. Think of it as split horizon between the routing table and the protocols, if that helps.
No, I've not answered your question directly, but I hope I've explained what the answer would be.
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