Chapter One: The Prohibition of Killing a Non-Jew

Point 6. There is nothing permitted to a Jew that is also prohibited to a non-Jew
This principle, that Jews are bound by the Seven Noahide Laws - is also found in the Talmud, Sanhedrin 59a:

There is nothing permitted to an Israelite yet forbidden to a non-Jew.

And this is exactly what is said in the Mechilta,

"And, after the reception of the Torah these restrictions were lessened? - That's a surprise!"

The Rambam emphasizes this principle, that a Jew is definitely obligated in the Seven Noahide laws. He says this on the point about the seven commandments (the beginning of Laws of the Kings, chapter 9):

Six precepts were commanded to Adam:

a) the prohibition against worship of false gods;

b) the prohibition against cursing God;

c) the prohibition against murder;

d) the prohibition against incest and adultery;

e) the prohibition against theft;

f) the command to establish laws and courts of justice.

Even though we have received all of these commands from Moses and, furthermore, they are concepts which intellect itself tends to accept, it appears from the Torah's words that Adam was commanded concerning them.
The prohibition against eating flesh from a living animal was added for Noah, as Genesis 9:4 states: 'Nevertheless, you may not eat flesh with its life, which is its blood.' Thus there are seven mitzvot.
These matters remained the same throughout the world until Abraham. When Abraham arose, in addition to these, he was commanded regarding circumcision. He also ordained the morning prayers.
Isaac separated tithes and ordained an additional prayer service before sunset. Jacob added the prohibition against eating the sciatic nerve. He also ordained the evening prayers. In Egypt, Amram was commanded regarding other mitzvot. Ultimately, Moses came and the Torah was completed by him.

One can say that the Seven Noahide Laws are the beginning of the Torah, and Moshe completed it through the reception of the Torah. And even though the laws might be nullified after the reception of the Torah, we were bound by them before. If so, just as before the reception of the Torah we were prohibitted to kill a non-Jew, so to are we prohibited from killing a non-Jew after the reception of the Torah.