Israeli Prime Minister Benjamin Netanyahu has instructed his cabinet ministers not to grant any interviews or express their own opinions in any public forum, so long as the regime in Cairo remains unstable.
"We are not relating at the moment to what is happening there, it is an Egyptian matter. We must worry about our own interests, and I am sure we are doing just that," Transport Minister Yisrael Katz told the Army Radio following news that Adli Mansour would be appointed interim President of Egypt.
"Morsi spent the last year just fitting his own people into the echelons of the regime," Ben-Eliezer said.
"The shake-up in Egypt will continue no matter who is elected, until Egypt returns to its secular base. It's not just the Muslim Brotherhood in Egypt that's the problem, but the entire movement - that hoped to take over the regime after 85 years," he added.
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"Interim President Adli Mansour is a man of the old regime, and I assume he will work quickly to abolish the constitution and bring about new elections," Ben-Eliezer said.
Channels of coordination between the Egyptian military and its intelligence and Israel's security forces continued to operate this week, despite the crisis, sources here said.
The two sides continued to cooperate with regard to stabilising the situation in Sinai and maintaining calm between Israel and Hamas along its border with the Gaza Strip.
Israel recently agreed to let Egypt deploy more troops in Sinai to prevent clashes with bands of Islamic extremists operating among the region's Bedouins.
Many analysts had questioned the fate of the peace treaty between the two countries in the face of ascendance of the Muslim Brotherhood regime under Morsi.
The landmark Egypt-Israel peace treaty was signed in 1979 by Egyptian President Anwar El Sadat and Israeli Prime Minister Menachem Begin following years of hostilities.