He wants evidence for the dhikrs that are said when bowing and prostrating
Question

What is the proof that we can say ‘subhaana rabbi al-azeem wa bi hamhdihi’ during rukoo and ‘subhaana rabbi al-a’la wa bi hamdhihi’ during sujood?.
Praise be to Allah.The dhikrs that are said when bowing and prostrating are of three types: 
1-Some are common to both and are said both when bowing and prostrating
2-Some are said only when bowing
3-Some are said only when prostrating.
The dhikrs that are common to both include the following: 
“Subhaanaka Allahumma Rabbana wa bihamdika Allahumma ighfir li (Glory and praise be to you O Allah, our Lord, O Allah forgive me).” 
It was narrated that ‘Aa’ishah (may Allah be pleased with her) said: The Prophet (peace and blessings of Allah be upon him) often used to say when bowing and prostrating: 
“Subhaanaka Allahumma Rabbana wa bihamdika Allahumma ighfir li (Glory and praise be to you O Allah, our Lord, O Allah forgive me).”
Narrated by al-Bukhaari (761) and Muslim (484). 
Another dhikr that is common to both is “Subbooh Quddoos Rabb ul-malaa’ikati wa’l-Rooh (Glorified, Holy, Lord of the angels and the Spirit)”. 
It was narrated from ‘Aa’ishah that the Messenger of Allah (peace and blessings of Allah be upon him) used to say when bowing and prostrating: “Subbooh Quddoos Rabb ul-malaa’ikati wa’l-Rooh (Glorified, Holy, Lord of the angels and the Spirit)”. Narrated by Muslim (487). 
Another common dhikr is: “Subhaana dhi’l-jabarooti wa’l-malakooti wa’l-kibriya’ wa’l-‘azamah (Glory be to the One Who has all power, dominion, magnificence and might).” 
It was narrated that ‘Awf ibn Maalik al-Ashja’i said: I prayed qiyaam with the Messenger of Allah (peace and blessings of Allah be upon him) one night. He stood and recited Soorat al-Baqarah, and he did not come to any verse that spoke of mercy but he paused and asked (for mercy). He did not come to any verse that spoke of punishment but he paused and sought refuge with Allah.  Then he bowed for as long as he had stood, saying whilst bowing: “Subhaana dhi’l-jabarooti wa’l-malakooti wa’l-kibriya’ wa’l-‘azamah (Glory be to the One Who has all power, dominion, magnificence and might).” Then he prostrated for as long as he had stood, and he said something similar when prostrating, then he stood and recited Aal ‘Imraan, then he recited soorah by soorah. 
Narrated by al-Nasaa’i (1132)and Abu Dawood (873). Classed as saheeh by Shaykh al-Albaani in Saheeh Abi Dawood. 
Among the dhikrs that are recited only in rukoo’ is: “Subhaana Rabbiy al-‘Azeem (Glory be to my Lord the Almighty).” 
It was narrated that Hudhayfah (may Allah be pleased with him) said: I prayed with the Prophet (peace and blessings of Allah be upon him) one night, and he started to recite al-Baqarah and I thought: he will bow when he reaches one hundred, but he carried on. Then I thought that he would finish it in the two rak’ahs, but he carried on. Then I thought he would bow after finishing it, but he started to recite al-Nisa’ and recited it all, then he started to recite Aal ‘Imraan and recited it all, reciting with a slow and measured pace. When he reached a verse that spoke of glorifying Allah, he glorified Allah; when he reached a verse that spoke of asking of Him, he asked of Him; when he reached a verse that spoke of seeking refuge with Him, he sought refuge with Him. Then he bowed and started saying, ‘Subhaana Rabbiy al-‘Azeem (Glory be to my Lord the Almighty).” And his bowing was almost as long as his standing. Then he said: “Sami’a Allahu liman hamidah (Allah hears those who praise Him).” Then he stood for a long time, almost as long as he had bowed. Then he prostrated and said, “Subhaan Rabbiy al-A’la (Glory be to my Lord Most High),” and his prostration was almost as long as his standing.
Narrated by Muslim (772). 
Among the dhikrs that are recited only when prostrating is: “Subhaana Rabbiy al-A’la (Glory be to my Lord Most High).” 
In the hadeeth of Hudhayfah quoted above it states that the Prophet (peace and blessings of Allah be upon him) used to say when prostrating: “Subhaana Rabbiy al-A’la (Glory be to my Lord Most High).” 
Abu Dawood (869) narrated that ‘Uqbah ibn ‘Aamir (may Allah be pleased with him) said: When the Messenger of Allah (peace and blessings of Allah be upon him) bowed he said:  “Subhaana Rabbiy al-‘Azeem wa bi hamdihi (Glory and praise be to my Lord the Almighty)” three times, and when he prostrated he said: “Subhaana Rabbiy al-A’la wa bi hamdihi (Glory and praise be to my Lord the Most High)” three times. 
The scholars differed as to whether the additional phrase bi hamdihi (praise) is saheeh or da’eef. The narrator of this phrase, Abu Dawood, said: We are afraid that this phrase may not be sound; it was narrated only by the people of Egypt. 
Shaykh al-Albaani classed it as saheeh in Sifat al-Salah (The Prophet’s Prayer Described) p. 146, then later on he classed it as da’eef in Da’eef Sunan Abi Dawood (1/338-340). 
It was also rejected by Ibn al-Salah and others, as stated in al-Talkhees al-Habeer (1/243). 
Ibn Qudaamah said in al-Mughni (1/297): Two reports are narrated from Imam Ahmad, one which says that he accepted it and another which says that he did not accept it. In the report that he did not accept it the reason give is the fact that the hadeeth without this additional phrase is more well-known. 
Ahmad ibn Hanbal was asked about it, according to a report narrated by Ibn al-Mundhir, and he said: As for me, I do not say bi hamdihi. 
And Allah knows best.