"To the bull"—this is the bull of the anointed priest. For what legal purpose? To make it indispensable. "To the bull"—this is the bull of the Day of Atonement. For what legal purpose? To make it indispensable. But that is obvious, for "a statute" is written regarding it! Rav Nachman said: it is needed only according to Rabbi Yehudah, who said that when "a statute" is written, it is written concerning the matters performed in the white garments inside, but as for the matters performed in the white garments outside, if one did a rite before its fellow, what he did is done. One might say: just as when out of order they are not indispensable, so too the sprinklings should not be indispensable; it teaches us otherwise. Rav Pappa objected: can you really say so? But it was taught, "And when he has finished atoning": if he atoned he finished, and if he did not atone he did not finish, the words of Rabbi Akiva. Rabbi Yehudah said: why should we not say, if he finished he atoned and if he did not finish he did not atone, that is, that if he omitted one of the applications he accomplished nothing? Rather, Rav Pappa said: it is needed only for "the" with the blood, and for the dipping. "The": Rav Acha bar Yaakov said it is needed only to permit the blood on the side of the finger. "With the blood": that there should be in the blood the measure of a dipping from the outset.
Which Rites of the Bull Truly Invalidate the Offering
Yalkut Shimoni on Torah 469:21
לַפָּר, זֶה פַּר כֹּהֵן מָשִׁיחַ. לְמַאי הִלְכְתָא, לְעַכֵּב. לְפַר, זֶה פַּר יוֹם הַכִּפּוּרִים, לְמַאי הִלְכְתָא, לְעַכֵּב, פְּשִׁיטָא "חֻקָּה" כְּתִיבָא בֵּיהּ. אָמַר רַב נַחְמָן, לֹא נִצְרְכָה אֶלָּא לְרַבִּי יְהוּדָה, דְּאָמַר כִּי כְּתִיבָא "חֻקָּה" אַדְּבָרִים הַנַּעֲשִׂין בְּבִגְדֵי לָבָן בִּפְנִים, אֲבָל דְּבָרִים הַנַּעֲשִׂין בְבִגְדֵי לָבָן בַּחוּץ הִקְדִּים מַעֲשֶׂה לַחֲבֵרוֹ מַה שֶּׁעָשָׂה עָשׂוּי, אֵימָא מִדִּכְסִדְרָן לָא מְעַכְּבָן הַזָּאוֹת נַמִּי לָא לִיעַכְּבוּ, קָא מַשְׁמַע לָן. מַתְקִיף לָה רַב פַּפָּא וּמִי מָצִית אָמַרְתְּ הָכִי, וְהָתַנְיָא "וְכִלָּה מִכַּפֵּר" אִם כִּפֵּר כִּלָּה אִם לֹא כִּפֵּר לֹא כִּלָּה, דִּבְרֵי רַבִּי עֲקִיבָא. אָמַר רַבִּי יְהוּדָה, מִפְּנֵי מָה לֹא נֹאמַר אִם כִּלָּה כִּפֵּר וְאִם לֹא כִּלָּה לֹא כִּפֵּר, שֶׁאִם חִסֵּר אַחַת מִן הַמַּתָּנוֹת לֹא עָשָׂה כְלוּם. אֶלָּא אָמַר רַב פַּפָּא, לֹא נִצְרְכָה אֶלָּא לְאֶת בַּדָּם וּבִטְבִילָה, "אֶת", אָמַר רַב אַחָא בַּר יַעֲקֹב, לָא נִצְרְכָה אֶלָּא לְהַכְשִׁיר אַמִין שֶׁבָּאֶצְבַּע, "בַּדָּם", שֶׁיְּהֵא בַּדָּם שִׁעוּר טְבִילָה מֵעִקָּרָא.