Sayyiduna Abu Sa'id (may Allah be pleased with him) said that the Prophet (peace and blessings be upon him) said: "When allegiance is pledged to two caliphs at one time, kill the latter of them."
Hadith Referenceالسنن الكبرى للبيهقي / كتاب قتال أهل البغي / 16547
Abu Hurayrah (may Allah be pleased with him) reports from the Prophet (peace and blessings be upon him): "Among the Children of Israel there was a succession of prophets: when one passed away, another came. But there will be no prophet after me; however, there will be many caliphs." They asked: "Then what do you command us?" He said: "Obey whichever of them comes first and give him his right; as for the rest, Allah will ask them about those placed in their care."
Hadith Referenceالسنن الكبرى للبيهقي / كتاب قتال أهل البغي / 16548
Sayyiduna Salim bin Ubayd (may Allah be pleased with him), who was one of the People of the Suffah, said: Sayyiduna Abu Bakr (may Allah be pleased with him) was with the Prophet (peace and blessings be upon him). It was said: "O companion of the Prophet (peace and blessings be upon him)! Has the Prophet (peace and blessings be upon him) passed away?" He said: "Yes," and they came to know it. Then he said: "Leave your companion to his cousins," meaning for washing him. Then he came out, and the Muhajirun gathered and began consulting, and they said: "Let us go to our Ansari brothers; we will consult them as well, for they too have a right (in this)." They came to the Ansar, and a man of the Ansar said: "One leader from you and one from us." Sayyiduna Umar (may Allah be pleased with him) said: "Two swords in one sheath; this will not work." He took the hand of Sayyiduna Abu Bakr (may Allah be pleased with him) and asked: "About whom are these three verses: 'When the two were in the cave' [Surah al-Tawbah: 40], 'When he said to his companion' [Surah al-Tawbah: 40], and 'Do not grieve; indeed, Allah is with us' [Surah al-Tawbah: 40]?" Then Sayyiduna Umar (may Allah be pleased with him) spread out the hand of Sayyiduna Abu Bakr (may Allah be pleased with him), pledged allegiance and told the people as well, and the people pledged allegiance in the finest manner.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب قتال أهل البغي / 16549
Ibn Ishaq relates the sermon of Sayyiduna Abu Bakr (may Allah be pleased with him) on that day, in which he (may Allah be pleased with him) said: It is not permissible for the Muslims to have two leaders, for then their commands and judgements would differ, division would spread among the Muslims and they would dispute among themselves; the Sunnah would be abandoned, innovation would appear, and trials would rise up, and there is no good in that for anyone.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب قتال أهل البغي / 16550