When should the words al-salaatu khayrun min al-nawm (prayer is better than sleep) be said? In the first adhaan or in the second?
Question

I have been studying with this Shaikh in Dubai for the last two years. He revealed an astonishing so called fact that `As salatu Khairum minan Naoom`in FAJR Adhan is a Bi`da cause Hazarat Bilal use to use these words in TAHAJJUD Adhan and IBN MAKTOUM use to call the FAJR Adhan where by he did not use these words.The other evidence is that, by meaning of these words you are trying to compare SLEEP with FAJR Salat which should not be the case. He also said that Shaikh Albaani had made a Du`aa for the youngsters of today asking ALLAH to give them Hidaya to stop this Bi`da. If my Teacher is correct, why is it being followed in Mecca and Madina as well. Please give me your comments.
Praise be to Allah.It is narrated in a number of saheeh ahaadeeth that the words al-salaatu khayrun min al-nawm (prayer is better than sleep) are to be said in the adhaan for Fajr. It is mentioned in some of them that this phrase is to be said in the first adhaan without stating what is meant by the phrase “the first adhaan” – is it the adhaan that is given before Fajr or is it the adhaan of Fajr itself? 
These ahaadeeth include the following: 
1 – It was narrated that Abu Mahdhoorah (may Allaah be pleased with him) said: I used to give the adhaan for the Messenger of Allaah (peace and blessings of Allaah be upon him) and in the first adhaan of Fajr I used to say: “Hayya ‘ala al-falaah, al-salaatu khayrun min al-nawm, al-salaatu khayrun min al-nawm, Allaahu akbar Allaahu akbar, laa ilaaha ill-Allaah (come to prosperity, prayer is better than sleep, prayer is better than sleep, Allaah is Most Great, Allaah is Most Great, there is no god but Allaah).” 
Narrated by Abu Dawood, 500; al-Nasaa’i, 647; classed as saheeh by al-Albaani in Saheeh Abi Dawood. 
2 – It was narrated that Ibn ‘Umar (may Allaah be pleased with him) said: In the first adhaan after the word al-falaah it said: Al-salaatu khayrun min al-nawm, al-salaatu khayrun min al-nawm (prayer is better than sleep, prayer is better than sleep).
 Shaykh al-Albaani said: 
It was narrated by al-Tahhaawi (1/82) with a hasan isnaad as al-Haafiz said in al-Talkhees (3/169). 
Al-Thamr al-Mustataab, p. 131 
These ahaadeeth provide evidence for those who said that this should be said in the first adhaan which comes at the end of the night. But the correct view is that it should be in said the adhaan which comes after the time for prayer has begun. That is for a number of reasons: 
(a)
The word awwal (first) is in relation to the iqaamah, and the iqaamah is the second call to prayer. In the saheeh Sunnah there are reports in which the iqaamah is called an “adhaan”, as in the words of the Prophet (peace and blessings of Allaah be upon him): “Between each two calls (adhaanayn)...” Narrated by al-Bukhaari, 598; Muslim, 838. 
In Saheeh Muslim (739) the adhaan which comes after the time for prayer begins is called the first adhaan. This appears in the hadeeth narrated by ‘Aa’ishah (may Allaah be pleased with her) about the prayer of the Messenger of Allaah (peace and blessings of Allaah be upon him). She said: He used to sleep during the first part of the night and wake up during the latter part, then if he had any need for intimacy with his wife he would satisfy that need, then he would sleep. Then when it was the time of the first call, he would wake up and pour water over himself, and if he was not junub he would do wudoo’ as a man does for prayer, then he would pray two rak’ahs. 
What is meant by two rak'ahs here is the regular Sunnah of Fajr, as stated by al-Nawawi in Sharh Muslim. 
(b)
In some saheeh ahaadeeth it is clearly stated that the adhaan in which this phrase appears is called adhaan salaat al-subh, adhaan al-fajr, etc. These phrases indicate that this adhaan should come after the time for the prayer begins and the adhaan that is given at the end of the night comes before the time for prayer begins. These ahaadeeth include the following: 
1 – It was narrated that Abu Mahdhoorah (may Allaah be pleased with him) said: I said: “O Messenger of Allaah, teach me the Sunnah of adhaan.” He wiped the front of my head and said: “Say Allaah akbar, Allaahu akbar, Allaah akbar, Allaahu akbar (Allaah is Most Great, Allaah is Most Great, Allaah is Most Great, Allaah is Most Great)… And if it is Fajr prayer, then say: Al-salaatu khayrun min al-nawm, al-salaatu khayrun min al-nawm (Prayer is better than sleep, prayer is better than sleep). 
According to another, similar report, it says: “Al-salaatu khayrun min al-nawm, al-salaatu khayrun min al-nawm (Prayer is better than sleep, prayer is better than sleep) in the first adhaan of dawn.” 
Narrated by Abu Dawood, 501; al-Nasaa’i, 633; classed as saheeh by al-Albaani in Saheeh Abi Dawood. 
According to another report narrated by Abu Dawood (504) from Abu Mahdhoorah (may Allaah be pleased with him): He used to say in Fajr: “al-salaatu khayrun min al-nawm (Prayer is better than sleep). Classed as saheeh by Shaykh al-Albaani in Saheeh Abi Dawood. 
2 – It was narrated that Anas (may Allaah be pleased with him) said: It is Sunnah for the muezzin, after saying Hayya ‘ala al-falaah (come to prosperity), to say: “al-salaatu khayrun min al-nawm (prayer is better than sleep)” twice. 
Shaykh al-Albaani (may Allaah have mercy on him) said: 
This was narrated by al-Daaraqutni (90), Ibn Khuzaymah in his Saheeh and al-Bayhaqi in his Sunan (1/423), and he said: its isnaad is saheeh. 
It was also narrated by al-Daraaqutni and al-Tahhaawi (1/82) via Hushaym from Ibn ‘Awn with the wording: In the adhaan for Fajr prayer, when the muezzin said: Hayya ‘ala al-falaah (come to prosperity), he said: “al-salaatu khayrun min al-nawm (prayer is better than sleep),” twice. This version was narrated by Ibn al-Sakan who classed it as saheeh, as it says in al-Talkhees, 3/148 
Al-Thamr al-Mustataab, p. 132 
These ahaadeeth show that this phrase appears in the adhaan for Fajr prayer. 
The adhaan which is the call to prayer is that which comes after the time for the prayer has begun, because the Prophet (peace and blessings of Allaah be upon him) said: “When the time for prayer begins, then let one of you give the adhaan for you.” Agreed upon. 
As for the adhaan which comes at the end of the night, this is not an adhaan for Fajr prayer as such, rather it is done “so that the one who is praying qiyaam may go back and the one who is asleep may wake up”, as is narrated from the Prophet (peace and blessings of Allaah be upon him) in al-Saheehayn. Thus it is clear that the phrase which appears in the adhaan which comes after the time for the prayer begins is not a bid’ah (innovation), rather it is Sunnah. 
The scholars of the Standing Committee were asked: 
What is the reason why the Sunnah of the Prophet (peace and blessings of Allaah be upon him), to say al-salaatu khayrun min al-nawm, is not done in the first adhaan for Fajr, as is mentioned in Sunan al-Nasaa’i, Ibn Khuzaymah and al-Bayhaqi? 
They replied: 
Yes, the this phrase should be mentioned in the first adhaan of Fajr, in accordance with the command of the Prophet (peace and blessings of Allaah be upon him). It is clear from the hadeeth that it is the adhaan that is done when the true dawn comes, and it is called ‘first’ in relation to the iqaamah, which is an adhaan according to sharee’ah, as it says in the hadeeth, “Between every two calls (adhaanayn) there should be a prayer.” The word adhaan here does not refer to the call that is given before the appearance of the true dawn. That call is prescribed in order to wake those who are asleep and so that those who are praying qiyaam may go back. It is not a call to announce the Fajr prayer. Whoever studies the ahaadeeth about this phrase will see clearly that they refer to the adhaan given to announce the time of Fajr, not the adhaan which is given at night just before dawn. End quote. 
Shaykh ‘Abd al-‘Azeez ibn ‘Abd-Allaah ibn Baaz, Shaykh ‘Abd al-Razzaaq ‘Afeefi, Shaykh ‘Abd-Allaah ibn Ghadyaan, Shaykh ‘Abd-Allaah ibn Qa’ood. 
Fataawa al-Lajnah al-Daa’imah li’l-Buhooth al-‘Ilmiyyah wa’l-Ifta’, 6/63 
For more information on the refutation of those who say that this phrase should be said in the adhaan which comes before the time for the prayer begins, please see al-Sharh al-Mumti’ by Shaykh al-‘Uthaymeen (may Allaah have mercy on him), 2/61-64 
As for the view of your teacher, that this phrase is comparing sleep with Fajr prayer, this is not correct, because this phrase is stating that prayer is better than sleep and thus encouraging the sleeper to stop sleeping and get up to do something that is better. 
And Allaah knows best.