Or HaChaim on Exodus 12:51/אור החיים על שמות י״ב:נ״א

1 ויהי בעצם היום הזה, It was on the self-same day, etc. The reason the Torah records the Exodus immediately after concluding the legislation of the Passover ceremonies is to remind us that the fulfilment of this מצוה is what triggered the redemption from Egypt. The Israelites had no other credentials than performance of this commandment.