> Permission is hereby granted... to deal in the Software without restriction, including without limitation the rights to... sublicense
Because you have the right to "sublicense", you can take MIT code and license it under another license (GPL) even though you are not the copyright holder.
The BSD license is less clear on this point (in fact this is one of the important differences) and whether the BSD's grant includes the right of sublicense is more debatable, but in my view the answer is "probably".
I am not a lawyer; I am not your lawyer; this is not legal advice.
The BSD license does not grant any sublicensing rights. It clearly says that the notice and disclaimers must be preserved.
The original license required copyright notices in binary code. A newer form does not, but this is for pragmatic reasons. The binary code is still copyrighted. If you decompile it into a source language, then you probably have to restore the copyright notice and disclaimers.
When a proprietary, binary-only program with a highly restrictive EULA contains BSD code, those sections of the machine language which correspond to the BSD code are not actually under the EULA. They are not sublicensed, but only used with permission, as granted by the original license.
> Permission is hereby granted... to deal in the Software without restriction, including without limitation the rights to... sublicense
Because you have the right to "sublicense", you can take MIT code and license it under another license (GPL) even though you are not the copyright holder.
The BSD license is less clear on this point (in fact this is one of the important differences) and whether the BSD's grant includes the right of sublicense is more debatable, but in my view the answer is "probably".
I am not a lawyer; I am not your lawyer; this is not legal advice.