International efforts to end Libya's civil strife and install a national unity government received a major boost on Tuesday as a rival administration that claimed control of the capital Tripoli said it was leaving office.

A statement issued by the Islamist-leaning National Salvation Government, whose militia allies have controlled Tripoli and most of western Libya since mid-2014, said its members had "ceased the functions entrusted to them" in order to avoid bloodshed and division.

A third government, backed by military strongman Khalifa Haftar, is still holding out in eastern Libya.

The news was hailed by UN Libya envoy Martin Kobler, who hours earlier was in Tripoli to meet Fayez Serraj, head of the UN-backed Presidency Council that is intended to serve as the nucleus of the national unity government.

"Good news. Now deeds must follow words," Kobler wrote on Twitter.

In another boost for the UN-backed peace plan, 70 members of the State Council, which is due to form a sort of upper house of parliament under the deal, met for the first time in Tripoli.

The body takes its membership from a former parliament which was revived by Islamist militias when they seized the capital in 2014, and its meeting will be seen as a sign of political backing for the unity deal in western Libya.

The unrecognized Tripoli government's hold on power has crumbled since Serraj's arrival in the capital six days ago, even though the latter was initially confined to the naval port where he came ashore.

It soon became clear that the Islamist-leaning administration did not enjoy enough support from the militias that have previously backed it to retain power.

UN and international efforts, aimed at weaning moderates in both camps away from hardliners and winning their backing for the peace deal, appear at last to have borne fruit in Tripoli.

It remains to be seen whether Serraj's embryonic Government of National Accord will be able to win allegiances in the east, where Haftar has much political as well as military influence. 

Resolving Libya's civil strife has become a key regional priority for European and international powers, alarmed by the growing presence of Islamic State jihadists.

The extremists have taken advantage of the chaos to carve out a territory for themselves in central Libya between the regions controlled by the rival governments.

European governments have also been spurred to action by people smugglers who have used the North African country's lawless coastline to offer refugees and migrants a risky sea crossing to Italy.

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