It should not be said that Allah سبحانه وتعالى has made some legal rulings specifically for the believers like the prayer such that they alone are addressed by it, so any address which begins with,
يا أيها الذين ءامنوا
"O you who believe,"
is specific for the Muslims, and what came general, like the trade and ribā, is for the Muslims and the disbelievers. This is incorrect because the objective of that which begins with 'O you who believe' is reminding [tadhkīr] the believers of their Imān. It is not a specification for the believers alone.
The evidence for this is Allah سبحانه وتعالى's saying:
يا أيها الذين ءامنوا كُتب عليكم القصاص
"O you who believe, qisās (equal retribution) has been prescribed on you..."
and it is established that the Messenger صلى الله عليه وسلم made the retribution for the killing upon the kuffār just as it is upon the Muslims without any difference, and also because the sayings of Allah سبحانه وتعالى:
لمن كان يرجوا الله واليوم الآخر
"...for him who hopes for Allah and the Last Day",
فردّوه إلى الله والرسول إن كنتم تؤمنون بالله واليوم الآخر
"Refer it back to Allah and the Messenger, if you believe in Allah and the Last Day..."
من كان يؤمن بالله واليوم الآخر
"He who believes in Allah and the last day..."
whose context indicate that they are a reminder to the believers about the (natural) requirement of Imān in Allah and the Last Day.
The first verse is,
لقد كان لكم في رسول الله أسوة حسنة لمن كان يرجوا الله واليوم الآخر
"Indeed in the Messenger of Allah, you have a good example for him who hopes for Allah and the Last Day."
The second verse is,
فإن تنازعتم في شيء فردّوه إلى الله والرسول إن كنتم تؤمنون بالله واليوم الآخر
"And if differ in a thing, refer it back to Allah and His Messenger if you believe in Allah and the Last Day."
The third verse is,
ذلكم يوعظ به من كان يؤمن بالله واليوم الآخر
"That will be an admonition given to him who believes in Allah and the Last Day."
All these verses are a reminder (and not a specification).
Likewise is the hadīth of the Messenger صلى الله عليه وسلم,
"Whosoever believes in Allah and the Last, then let him speak good or keep silent", narrated by al-Bukhāri, Muslim and others.
All of this is a reminder of the Imān, and not a condition for legal responsibility with respect to the legal rulings.
Therefore, the conjunction of the address with 'O you who believe' does not specify it for the Muslims, but it is a reminder to them of their belief. Accordingly, the address of legal responsibility remains general, including both the disbelievers and the Muslims. Hence the kuffār are addressed with the entirety of the Shari'ah, foundations (creed) as well as branches (jurisprudence), and the Khalifah is commanded to implement all the legal rulings upon them. Excluded from the implementation, but not the address, are those legal rulings whose non-implementation on the kuffār or whose specificity to the Muslims is established by a text of the Qur'an or Hadīth.
All other legal rulings are implemented upon the kuffar as they are implemented on the Muslims.
Source: The Islamic Personality Volume 3 by Sheikh Taqiuddin an-Nabhani.