based on Shirat Hageula, p. 19-21

In the Haggada, we read p’sukim (Yechezkel 16:6-7) that use the parable of the birth of a baby girl to represent Bnei Yisrael’s liberation from Egyptian bondage. The navi repeats the phrase "Through your blood shall you live." The midrash (Yalkut Shimoni 16) relates this to the blood of mila and the blood of Korban Pesach. What is the significance of the joint appearance of these two bloods?
Let us take a deep look at the liberation from Egypt, which was planned before bondage began (see Bereishit 15:13-14). We can understand the joy of those liberated from apparently random subjugation. However, when the subjugation and the liberation were decreed in advance, one could view the liberation as no more than the end of Hashem’s decree rather than a positive redemption. What, then, makes the Exodus from Egypt so special?
Avraham asked: "How will I know?" (ibid. 8). How does "You shall surely know …" (ibid. 13) answer his question? Avraham’s unique path started with him acting expansively with "the souls he made in Charan" and his concern for all, from the people of Sodom to nomads who visited his tent. But then he complained by doubting the significance of this legacy if he had no children (ibid. 3). Then mila also singled out his family as different from all others. Although the letter heh that was attached to his name confirms his remaining commitment to being "the father of many nations," his methods seem to have changed.
The unity of mankind is fundamental to the Torah. We all possess a Divine Image, and the One G-d of the world is ours jointly; he is not just a national G-d. But specifically due to this, there must be a leading nation, which serves as a heart among the body’s organs. With this in mind, Avraham asked how he should know he will merit this special task. Hashem answered that he will know because a special historical path awaits his descendants. Their experiences will train them to embrace a national legacy of tikkun olam.
The Sons of Israel entered Egyptian society with a belief that if they contributed to it, they would be seen as loyal citizens and treated as of the same blood. Once Yosef died, they stopped singling themselves by doing mila (Shemot Rabba 1:8) but learned the hard way that their feelings of unity went unreciprocated. While Bnei Yisrael stood out (Haggada), this was the Egyptians’ doing, as they "stamped ‘Jid’ on their passports" and put them in ghettos.
In some ways, many ethnic groups have been subjugated and have fought successfully for their freedom. However, everything having to do with our liberation was uniquely miraculous.
"Through your blood shall you live." There is blood that makes one recognize his uniqueness. However, this applies to others as well. There are non-Jews who have been circumcised, and some have had their blood spilled in pursuit of independence. However, they do so to have an easier life and gain access to physical desires, like a roasted goat. It is special, though, to take the roasted goat and turn it into a sacrifice – to become independent only to agree to be dependant on the Master of the Universe. Here is where the double blood comes in. We give of ourselves for the service of Hashem instead of taking natural treasures that Hashem gives for our own interests like the nations do. We share with the world the word of Hashem that emanates from Zion. We pray for a restoration of a Sanctuary like at the time of Moshe and Shlomo (see Vayikra Rabba 7) – the former, at a time of isolation and the latter, at a time of expansion. The convergence of the two is the content of our geula.