The clashes erupted after troops tried to expel rebels from a strategic position near Amran controling the road to Sanaa, local officials said.
The air force intervened to back the army and its allies from the Sunni Islah party, they said.
The rebels, known also as Ansarullah and backed by local tribes, retaliated by attacking telecom installations and blocked traffic on the main Amran-Sanaa road, the officials said.
"Seven soldiers were killed and 16 others wounded in the battle," a medical source told AFP.
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The rebel spokesman Mohammed Abdulsalam declined to give a death toll.
But he condemned the air force's intervention as a "very serious development" and warned of "consequences".
In Sanaa, unknown gunmen opened fire on the home of Prime Minister Mohamed Basindawa, without causing injury, a security source said.
The rebels are suspected of trying to enlarge their sphere of influence as Yemen is split into six regions, pushing out from their mountain strongholds in the far north to areas closer to Sanaa.
The rebels complained Yemen would be divided into rich and poor regions under a federalisation plan agreed in February following national talks as part of a political transition.
In February, they seized areas of Amran province in fighting with tribes that left more than 150 people dead.