The reformist alliance formed in support of moderate President Hassan Rouhani had campaigned against both Yazdi, the outgoing chairman of the Assembly of Experts and Mesbah-Yazdi, an outspoken opponent of the reformists.
But a third conservative, Ahmad Jannati, who had been targeted by the pro-Rouhani coalition, narrowly kept his seat on the assembly, a powerful committee which oversees the work of supreme leader Ayatollah Ali Khamenei, state television reported.
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Verification of the result is not expected for several days.
The election to the 88-member Assembly of Experts is important because it will pick Khamenei's successor if he dies during its eight-year term.
Rouhani placed third in polling for the assembly in Tehran and his key ally, former president Ayatollah Akbar Hashemi Rafsanjani took first place.
