Grading: This hadith is sahih and its chain is qawi (strong).
Technical note / Defect ('illah): The narrators of the chain are clarified as follows: Abu Qilabah refers to 'Abd al-Malik bin Muhammad bin 'Abd Allah al-Raqashi, Abu 'Asim refers to al-Dahhak bin Makhlad al-Nabil, and Ibn 'Ajlan refers to Muhammad bin 'Ajlan al-Madani.

Reference / Source: This narration was transmitted by Imam Ahmad bin Hanbal in his Musnad (13/ 7599) and (16/ 10707), Imam al-Tirmidhi (2746), Imam al-Nasa'i (9974) and Imam Ibn Hibban (2358) through various routes from Muhammad bin 'Ajlan, with this same chain.
Grading: Imam al-Tirmidhi declared this narration a "hasan hadith".

Reference / Source: Imam Ibn Majah (968) transmitted a similar narration through the route of 'Abd Allah bin Sa'id al-Maqburi, from his father (Sa'id bin Abi Sa'id).
Technical note / Defect ('illah): The narrator in this chain, 'Abd Allah bin Sa'id al-Maqburi, is "matruk" (one whose narration has been abandoned).

Note / Clarification: This hadith will come later at no. (7879) and (7880) through the route of Muhammad bin 'Abd al-Rahman bin Abi Dhi'b, from Sa'id al-Maqburi, in which Sa'id narrates from Abu Hurayrah (may Allah be pleased with him) by way of his father Kaysan (Abu Sa'id al-Maqburi); there the name of Kaysan is added.

Important point: The narration of Muhammad bin 'Ajlan is "mahfuz" (preserved).
Corroborating chains and supporting reports: Ibn 'Ajlan was corroborated by 'Abd al-Rahman bin Ishaq, as found in Abu Ya'la (6627) and Ibn Khuzaymah (922); Ibn Jurayj and Abu Ma'shar also supported him, as Imam al-Daraqutni mentioned in "al-'Ilal" (2056), although the father of Sa'id is not mentioned there. Nevertheless, Imam al-Daraqutni says that it appears Ibn Abi Dhi'b preserved it accurately.
Technical note / Defect ('illah): And Allah knows best, it apparently seems that Sa'id al-Maqburi used to narrate it from Abu Hurayrah (may Allah be pleased with him) in two ways (sometimes with the intermediary and sometimes without it). Further clarification of this will come in the takhrij of hadith (7879), that Ibn Abi Dhi'b himself narrated it from Sa'id al-Maqburi in two ways.

Details of the narration: Imam Ahmad (14/ 8631), Imam al-Bukhari (6224), Imam Abu Dawud (5033) and Imam al-Nasa'i (9989) narrated it as marfu' through the route of Abu Salih from Abu Hurayrah (may Allah be pleased with him): "When one of you sneezes, let him say 'al-hamdu lillah', and let his brother or companion say 'yarhamuk Allah'; then when he says 'yarhamuk Allah' to him, the one who sneezed should say 'yahdikum Allah wa yuslihu balakum' (may Allah guide you and set your affairs right)."

Reference / Source: It was narrated by Imam Ahmad (15/ 9162) and (16/ 10695), Imam Muslim (2994), Imam al-Tirmidhi (370) and Imam Ibn Hibban (2357) and (2359) as marfu' through the route of al-'Ala' bin 'Abd al-Rahman bin Ya'qub, from his father, from Abu Hurayrah: "Yawning is from Satan, so when one of you yawns, let him suppress it as much as he can."
Grading: Imam al-Tirmidhi declared it "hasan sahih".

General principle: The word "yushammituhu" has been narrated with both shin (with dots) and sin (without dots).
Note / Clarification: If it is read with shin ("al-tashmit"), it means: "May Allah keep away from you the gloating (shamatah) of enemies". And if it is read with sin ("al-tasmit"), it is derived from "al-samt", which means "the right path and guidance". This explanation was given by Imam al-Nawawi in "Sharh Muslim".