If the Prophet was saying the first tashahhud, he would get up for the third rak’ah with takbir.
If the Prophet (peace be upon him) was saying the first tashahhud, he would get up for the third rak’ah with takbir (saying Allahu Akbar). (Al-Bukhari and Muslim)
If he was at the last tashahhud, he would then say:
“O Allah, send prayers on Muhammad, and on the family of Muhammad, as you sent prayers on [Ibrahim, and on] the family of Ibrahim; You are indeed Worthy of Praise, Full of Glory. O Allah, send blessings on Muhammad, and on the family of Muhammad, as you sent blessings on [Ibrahim, and on] the family of Ibrahim; You are indeed Worthy of Praise, Full of Glory.” (Al-Bukhari and Muslim)
A closer look at the meanings:
Send prayers on Muhammad: You are asking Allah to exalt him and raise his rank, as noted by Ibn Hajar.
Send blessings on Muhammad: You are asking Allah to increase blessings for the Prophet, meaning to give the Prophet the good which Allah granted to Ibrahim (as) as well as its multiplying and increase.
The Prophet (peace be upon him) told us about the virtues of passing salams on him. He has said:
“If anyone of my community says one prayer on me, Allah blesses him ten times, ten good actions are written down for him and ten bad actions are effaced from him.” (An-Nasa’i)
After sending prayers on the Prophet:
When the Prophet heard a man glorifying and praising Allah, and sending prayers on the Prophet in prayer, he said to him: “Supplicate, and you will be answered; ask, and you will be given.” (An-Nasa’i)
Notice how the tashahhud follows the same sequence of the etiquette of getting our supplications answered: praising Allah, passing peace and blessings on His Prophet, and then we are given the opportunity to make du`aa’, which the Prophet has told us Allah will answer in sha’Allah.
He then told us what to say: “When one of you has finished the [last] tashahhud, he should seek refuge with Allah from four things saying:
“‘O Allah! I truly seek refuge with You from the punishment of the grave, and from the punishment of Hellfire, and from the trials of living and dying, and from the evil [trials] of Al-Dajjal.’ Then he should supplicate for himself with what occurs to him.” (Muslim)
If we have just met with someone, and are departing, we ask “Do you want anything from me?” Allah’s Mercy is infinitely more bountiful. At the end of our brief meeting, it is as though He is telling us “Is there anything else you want?” by granting us this final du`aa’.
The Tasleem:
Next, the Prophet (peace be upon him) would salute to his right saying:
“May Peace and Allah’s Mercy be on you and he would turn to his left and say the same.” (At-Tirmidhi)
After the prayer:
The Prophet would then seek forgiveness three times – we should do this to seek forgiveness for our shortcomings in prayers. There are many other du`aa’ to say, and they can be found here.
But there is something we should be wary of, and this was something the Prophet feared for us too. He said:
“If you did not sin, I would have feared for you what is greater; and that is self-admiration.” (Al-Bayhaqi)
Meaning, now that we have improved our prayer in sha’Allah and have a heightened khushu`, we should not regard ourselves as better than others. Ibn Al-Qayim has said that self-admiration even invalidates our acts! We should remember all of those past prayers that we prayed badly, and know that whatever good we are able to do, it is from Allah, as He has said in the Qur’an:
And whatever you have of favor – it is from Allah… (An-Nahl 16:53)
And all this that we have said, is not even a drop in the ocean in relation to the secrets and pleasures of prayer.
May we put into practice all that we learned, and regard each prayer as a sacred, beautiful meeting with Allah.
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Source: suhaibwebb.com.