JERUSALEM — After Mohamed Morsi, an Islamist, was elected president of Egypt a year ago, he refused any contact with Israelis, raising deep anxiety here and concern about the future of the Israel-Egypt peace treaty, a cornerstone of regional stability for decades.

But with Mr. Morsi’s ouster and the crackdown against the Muslim Brotherhood in Egypt this week, Israelis see the prospect of a return to what they view as a more reliable status quo, as well as a weakening of Hamas, the militant Islamic group that runs Gaza.

And yet, the good news for Israel remains tempered by the danger of chronic instability next door.

“What is important for Israel is a stable Egypt,” said Shaul Shay, a former deputy head of Israel’s National Security Council. “I don’t see the Muslim Brotherhood there swallowing the blow and waiting another 80 years to try to return to power. The story is not over, despite the fireworks in Cairo.”

While Mr. Morsi served as head of state, Israel’s only line of communication with Cairo was through the Egyptian military and security establishment, which is now controlling Egypt’s political process. Perhaps more reassuring to Israel is the role of Gen. Abdul-Fattah el-Sisi, the top commander who led the move to depose Mr. Morsi.