Regarding matzah, the verses talk of two mitzvot: (a) to eat matzah and marror with the korban pesach (Shemot 12:8) and (b) to eat matzah for all seven days (Shemot 12:8; 13:7). The independence of these two mitzvot from one another can be seen by the fact that the pesach sheni, the make-up pesach sacrifice brough a month later, was to be eaten with matzah and marror (Bamidbar 9:11) even though one can eat chametz and has no other mitzvah to eat matzah at that time. There is one set of verses that implicitly connects these two mitzvot:
You shall therefore sacrifice the Passover to the Lord your God, of the flock and the herd... You shall eat no chametz with it; seven days shall you eat matzah with it, the bread of affliction; for you came out of the land of Egypt in haste...
In these verses the seven days of eating matzah is connected to the eating on matzah together with the korban pesach - "seven days you shall eat matzah with it," i.e., with the korban pesach. Another way of reading this verse is not a connecting of the seven days of matzah to the first night, but a collapsing of those seven days into the eating of the first night. The Mekhilta of Rabban Shimon ben Yochai (12:18) picks up on this, and uses these words - "on it" to teach that the mitzvah of matzah applies only on the first night. However, those words also imply that matzah is eaten only together with the korban pesach. The Mekhilta in the end strikes a balance between the verses - the verse "on it" teaches that the mitzvah is the first night, and the verse "seven days you shall eat matzah" teaches that it is an independent mitzvah that night, not connected to the korban pesach.
Thus, matzah is established as a mitzvah independent of the korban pesach. However, a middle position is possible - that matzah can be eaten separately from the korban pesach but only when there is a Beit HaMikdash, so it is connected in a virtual way to the korban pesach. The Mekhilta suggests and in the end rejects this, quoting a snippet of a verse: "'In the evening you shall eat matzah' (Shemot 12:18) - the verse has established it as an independent obligation." This is clearly not the simple sense of the verse. The verse states "On the fourteenth of the month in the evening you shall eat matzah until the twenty-first of the month in the evening." However, but focusing a few words in the middle, Hazal established the independent identity of the mitzvah of matzah.
It is worth noting that there were those who disagreed, and who were of the opinion that matzah exists as a mitzvah only together with the korban pesach. The Gemara (Pesachim 120a) quotes the opinion of Rav Acha bar Yaakov that matzah nowadays is only a rabbinic mitzvah, since there is no korban pesach. Similarly, the Tosefta (Pesachim 2:22), views matzah and marror only as part of the korban pesach, and quotes R. Shimon who exempts women from all of them, since - according to him - women are exempt from the korban pesach. [The Gemara (Pesachim 91b) reinterprets this statement in line with its position that matzah is an independent mitzvah.]
The Tosefta ends by stating that matzah, marror, and the korban pesach can be done separate of one another. This does not mean that they are independent mitzvot. Rather, this is the middle position rejected by the Mekhilta. At the time of the Temple, when there is a korban pesach, the mitzvah of matzah can be done even if one is not eating the korban pesach since at least it is "on it," and connected in some virtual way to the korban. This position is also that of a braitta, which states that when a man is uncircumcised, although he cannot eat the korban pesach, he still eats matzah and marror. The inclusion of marror (which no one suggests is an independent mitzvah) and the need to derive from a verse that one still eats matzah and marror, make it clear that these mitzvot exists only in connection with the korban pesach, and are not fully independent mitzvot.
Now, together with establishing the mitzvah of matzah as an independent mitzvah of the seder night, also came the transforming of the mitzvah of the seven days of eating matzah. This mitzvah, mentioned several times in the Torah, was understood by Hazal to be an option, not an obligation. Indeed, if there were a general obligation to eat matzah all seven days, there would be little need to have a separate mitzvah to eat it that night. Thus, in the Gemara (Pesachim 120a), the gemara quotes a braitta which first establishes that there is no mitzvah to eat matzah all seven days (based on the fact that one verse only states six days, not seven days, of eating), and then goes on to establish that there is a mitzvah to eat it on the seder night. We have thus fully collapsed the seven day mitzvah into a mitzvah of the first night.
This collapsing of the mitzvah of matzah into the first night is nothing less than the transformation of Chag haMatzot into Chag haPesach. The Torah clearly recognizes two periods, "Pesach" - the 14th of Nissan, the time of the brining of the korban pesach (Bamidbar 33:3, and see Tosafot Rosh Hashana (13a), s.v. di'akrivu), and "Chag HaMatzot" - the seven day period from the 15th of Nissan through the 21st of Nissan (Shemot 23:13; 34:10. And see Vayikra 23:5-6 which juxtaposes the two). Now, according to the simple sense of the verses, Pesach is celebrated by the bringing of the korban pesach, and Chag HaMatzot is celebrated by eating matzot all seven days. This exactly parallels Chag HaSukkot, which is celebrated by sitting in a sukkah all seven days. Chag HaMatzot, thus, should have significance independant of the seder night. But by focusing the mitzvah of matzah of the first night, we have eating of matzah for the duration of Pesach into an extension of the real mitzvah, the eating of matzah on the seder night [and thus, in a way, fulfilled the verse "seven days you shall eat on it matzot."]. We have made the focus of the chag to be the seder night, and have transformed Chag haMatzot into Pesach.
In this way, also, the korban pesach has been replaced by the mitzvah of matzah. Thus, whereas in the Torah the mitzvot of the night centered around the korban pesach, for us, the mitzvot of the night - and in particular, the other mitzvah d'oraitta of sippur yitziyat mitzrayim, the telling of the story of the Exodus - center around the matzah. Take the statement of Rabban Gamliel in the mishna (Pesachim 116b) that one who does not say "pesach, matzah, and marror," that is - who does not connect the story to the mitzvot of pesach, matzah, and marror - does not fulfill his or her obligation. This puts all the mitzvot on equal footing, and implicitly highlights the korban pesach, to which matzah and marror are attached. However, today, when we say this and explain the significance of these foods, we do not lift up or even point to the shank bone, lest one suspect we are bringing sacrifices outside of the Temple (Pesachim 116b). The focus of the hagaddah and the mitzvah of sippur yitziyat mitzrayim, then, naturally shifts to mitzvah of matzah.
The hagaddah's connection to, and even dependency on, the mitzvah of matzah is expressed halakhically as well. The Gemara (Pesachim 116b) states that the mitzvah of sippur yitziyat mitzrayim is dependent on the mitzvah of matzah. If matzah is biblical, so is the mitzvah of sippur yitziyat mitzrayim, and if matzah is rabbinic, then there is only a rabbinic mitzvah to say the hagaddah. This is learned from the verse "because of this" (Shemot 13:8) - ba'avor zeh - "'this'," says Rava, "means because of matzah and marror." What is noticeably absent is the mention of the korban pesach. The mitzvah of the hagaddah survives because we have connected it to the mitzvah of matzah and not to the mitzvah of the korban pesach. In other words, because matzah has taken the place of the korban pesach. This is made clear in the passage in the hagaddah (from the Mekhilta):
Perhaps from Rosh Chodesh? The verse teaches, "on that day." If "on that day," (Shemot 13:8) perhaps from the day before? The verse teaches, "because of this." "Because of this" I only said at a time when matzah and marror are present before you.
Why would I have said "the day before"? Because it was the day of the bringing of the korban pesach. And, indeed, one of the passages of the mitzvah of sippur yitziyat mitzrayim is explicitly connected to the bringing of the korban pesach (Shemot 12:25-27). Nevertheless, we learn from this verse that the mitzvah of the hagaddah is linked to the matzah, and thus still is applicable today. Matzah is the focus, not the korban pesach.
This focus on matzah, and the connection of the hagaddah to it, is made complete by the statement of Shmuel:
Shmuel said: "'Bread of affliction'" (Devarim 15:3) - bread that one says over it many things." We taught similarly: ""Bread of affliction' - bread that one says over it many things. Another interpretation: 'Bread of affliction' - it is written 'poor person' - what is the manner of a poor person? With a broken piece. Here too with a broken piece."
The matzah is the bread that we say many things over - it is the focal point of the hagaddah. It is for this reason that we begin the hagaddah with "ha lachma anya" - this is the bread of affliction, this is the bread over which the hagaddah will be said. [Notice also the end of that passage - yasei vi'yifasch, let him come and eat/celebrate the Pesach - implicitly identifying the matzah with the korban pesach.] And thus, at yachatz, we break the matzah right before magid, so this bread over which we say the hagaddah will also be lechem ani, a poor person's bread, a broken piece of bread.
In the absence of the korban pesach, the mitzvah of matzah moved to the forefront. It was understood to be an independent mitzvah, and took the place of the korban pesach as the centerpiece of the seder. The entire hagaddah now revolves around the matzah, the lechem oni/lechem ani. In the absence of the korban pesach, rather than shifting our attention to the seven days of Chag HaMatzot, we have continued to focus on our attention of the seder night, and Chag HaMatzot has been transformed into Chag haPesach.