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The pasuk tells us that Moshe brought Yosef's bones because Yosef made his brother's swear an oath that they would bring his bones with them when Gd took them out of Mitsrayim. Rashi tells us that when the Bnai Yisrael left Mitrayim, they simultaneously brought the bones of the rest of the shevatim as well. In fact, Rashi says that Yosef took it as a given(1) that the bones of the shevatim would be taken out.
The Chizkuni and Siftei Hachamim (and others) point out that Yosef's language was peculiar: "השבע השביע" is seemingly, unnecessarily repetitive. They explain that the repetition comes to teach us that Yosef not only made his brothers swear to take his bones to Israel, but to make each successive generation swear as well. So, this obligation (of taking Yosef's bones from mitsrayim to Israel) was a binding obligation on each generation until dor yetsia.
This obligation is possibly the key to understanding why the Torah only mentions that Moshe brought Yosef's bones but did not mention the other members of the Bnai Yisrael who brought the bones of the other shevatim. This appears to be the first time that we are introduced to R' Hanina's principle "גדול המצווה ועושה יותר ממי שאינו מצווה ועושה" (One who is commanded and performs a mitzva is greater than one who is not commanded and performs it.) The Bnai Yisrael voluntarily brought out the bones of each of the shevatim; and while this was certainly a commendable act, it was not as great as Moshe bringing out the bones of Yosef, whom the Bnai Yisrael were obligated to bring to Israel on an oath made at his bedside and passed down each generation until the Exodus, was the greater act, and thus warranted mention in the Torah.
(1) Perhaps Yosef expected that each shevet would want to bring the bones of its namesake and bury them in their tribal land. Because Yosef's shevet was split into two, he may have been concerned that his sons' shevatim would have been content to have their respective namesakes and would not have bothered with his bones? I did not find an answer to this question.
The pasuk tells us that Moshe brought Yosef's bones because Yosef made his brother's swear an oath that they would bring his bones with them when Gd took them out of Mitsrayim. Rashi tells us that when the Bnai Yisrael left Mitrayim, they simultaneously brought the bones of the rest of the shevatim as well. In fact, Rashi says that Yosef took it as a given(1) that the bones of the shevatim would be taken out.
The Chizkuni and Siftei Hachamim (and others) point out that Yosef's language was peculiar: "השבע השביע" is seemingly, unnecessarily repetitive. They explain that the repetition comes to teach us that Yosef not only made his brothers swear to take his bones to Israel, but to make each successive generation swear as well. So, this obligation (of taking Yosef's bones from mitsrayim to Israel) was a binding obligation on each generation until dor yetsia.
This obligation is possibly the key to understanding why the Torah only mentions that Moshe brought Yosef's bones but did not mention the other members of the Bnai Yisrael who brought the bones of the other shevatim. This appears to be the first time that we are introduced to R' Hanina's principle "גדול המצווה ועושה יותר ממי שאינו מצווה ועושה" (One who is commanded and performs a mitzva is greater than one who is not commanded and performs it.) The Bnai Yisrael voluntarily brought out the bones of each of the shevatim; and while this was certainly a commendable act, it was not as great as Moshe bringing out the bones of Yosef, whom the Bnai Yisrael were obligated to bring to Israel on an oath made at his bedside and passed down each generation until the Exodus, was the greater act, and thus warranted mention in the Torah.
(1) Perhaps Yosef expected that each shevet would want to bring the bones of its namesake and bury them in their tribal land. Because Yosef's shevet was split into two, he may have been concerned that his sons' shevatim would have been content to have their respective namesakes and would not have bothered with his bones? I did not find an answer to this question.
וַיִּקַּ֥ח מֹשֶׁ֛ה אֶת־עַצְמ֥וֹת יוֹסֵ֖ף עִמּ֑וֹ כִּי֩ הַשְׁבֵּ֨עַ הִשְׁבִּ֜יעַ אֶת־בְּנֵ֤י יִשְׂרָאֵל֙ לֵאמֹ֔ר פָּקֹ֨ד יִפְקֹ֤ד אֱלֹהִים֙ אֶתְכֶ֔ם וְהַעֲלִיתֶ֧ם אֶת־עַצְמֹתַ֛י מִזֶּ֖ה אִתְּכֶֽם׃
And Moses took with him the bones of Joseph, who had exacted an oath from the children of Israel, saying, “God will be sure to take notice of you: then you shall carry up my bones from here with you.”
והעליתם את עצמתי מזה אתכם. לְאֶחָיו הִשְׁבִּיעַ כֵּן, לִמְּדָנוּ שֶׁאַף עַצְמוֹת כָּל הַשְּׁבָטִים הֶעֱלוּ עִמָּהֶם שֶׁנֶּאֱמַר אִתְּכֶם (מכילתא):
והעליתם את עצמתי מזה אתכם THEN YE SHALL BRING UP MY BONES AWAY HENCE WITH YOU — It was his brothers whom he made swear thus; consequently this tells us that they (the Israelites who left Egypt) must have brought up with them the bones of all Jacob’s sons (lit., the tribes) since it is said “with you” (with your own bones) (Mekhilta d'Rabbi Yishmael 13:19:11).
השבע השביע. הִשְׁבִּיעָם שֶׁיַּשְׁבִּיעוּ לִבְנֵיהֶם. וְלָמָּה לֹא הִשְׁבִּיעַ בָּנָיו שֶׁיִּשָּׂאוּהוּ לְאֶרֶץ כְּנַעַן מִיָּד כְּמוֹ שֶׁהִשְׁבִּיעַ יַעֲקֹב? אָמַר יוֹסֵף, אֲנִי שַׁלִּיט הָיִיתִי בְמִצְרַיִם וְהָיָה סֵפֶק בְּיָדִי לַעֲשׂוֹת, אֲבָל בָנַי לֹא יַנִּיחוּם מִצְרַיִם לַעֲשׂוֹת, לְכָךְ הִשְׁבִּיעָם לִכְשֶׁיִּגָּאֲלוּ וְיֵצְאוּ מִשָּׁם שֶׁיִּשָּׂאוּהוּ:
השבע השביע HE HAD STRAITLY ADJURED — He had made them swear that they would make their children swear. But why did he not make his sons swear that they would take him to the land of Canaan immediately on his death, as Jacob had made his sons swear? Joseph said: I was a ruler in Egypt and I had sufficient power to do what my father desired; but my sons — the Egyptians will not permit them to do this. Consequently he made them swear that when they would be delivered and would go forth from there that they would take him up with them (Mekhilta d'Rabbi Yishmael 13:19:8).
כי השבע השביע שבועה אחר שבועה השביע את אחיו שישביעו את בניהם דור אחר דור, אמר להם אין אני מטריח אתכם שלא כדין, משכם גנבתוני כדכתיב ויבא שכמה, לשכם החזרוני וכן עשו דכתיב ואת עצמות יוסף אשר העלו ממצרים קברו בשכם.
כי השבע השביע, the repetition of the words “he made swear repeatedly,” was that Joseph did not know which generation would experience the Exodus. Joseph had told his brothers that by committing them to this oath he was not asking them for a favour: they who had been instrumental in his being dragged to Egypt as a slave if they wanted to complete the act of repentance,must ensure that he would be brought to burial in the land of his forefathers from where he had been kidnapped. He had performed a mission on his father’s behalf when he came to Sh’chem (Genesis 37,14) and this is why he expected to be buried near there. This is where he was interred eventually, as we know from Joshua 24,32.
השביעם שישבעו לבניהם. דליכא למימר דברה תורה כלשון בני האדם, וכי עלתה על דעת שאותו הדור יעלו ממצרים, והלא היה להם להיות במצרים ארבע מאות שנים, ולפחות רד"ו שנים, ועוד, מאי נתינת טעם הוא. אלא שהשביעם וכו' ודלא כהרא"ם. [נחלת יעקב]:
Swear that they will make their children swear. [How does Rashi know this? The answer is:] It cannot be that [the repetitive השבע השביע is merely because] the Torah speaks as people do, [and the oath was for Yoseif’s generation alone]. For how is it possible that Yoseif’s generation would leave Egypt? They needed to be in Egypt for 400 years, or at least for 210! Furthermore, [if Yoseif’s oath was for that generation alone,] how would [we explain the word כי ? I.e., how would] the oath be the reason [for Moshe to take along Yoseif’s bones]? Perforce, “He made them swear that they will make their children swear.” This is preferable to Re”m’s explanation. (Nachalas Yaakov)
אמרי אין מקבלים עליהן שכר כמצווה ועושה אלא כמי שאינו מצווה ועושה דא"ר חנינא גדול המצווה ועושה יותר ממי שאינו מצווה ועושה:
The Sages said in response: Rav Yosef meant that they do not receive the reward as does one who is commanded to perform a mitzva and performs it, but as does one who is not commanded to perform a mitzva and performs it anyway. As Rabbi Ḥanina says: One who is commanded and performs a mitzva is greater than one who is not commanded and performs it.