Yehoshua was born 8 August 1931, in Jerusalem’s Mishkenot Shaananim. His family was of Spanish origins but had lived in the Land of Israel for thirteen generations. His father was Shammo Matza, who sold textiles, and his mother was Yona née Parnas. There were four brothers in the family, and they all joined underground movements at some point.
At a young age, he joined Beitar and Young Maccabee. At thirteen, when he was still a student in the Alliance School, he joined Lehi. At first, he was a “paperboy,” i.e. he was involved in putting up posters and disseminating promotional materials. Later, he was trained on light weapons and pistols in a course which took place in the Bukharan neighborhood; he then took part in combat operations. As the War of Independence began, he served in the Lehi camps in Jerusalem, Camp Dror and Camp Eldad, and he participated in defensive and offensive operations in the city, such as breaching the walls at the New Gate, the battle for Notre Dame (“the roof of death”) and the capture of Der Yassin. After the Bernadotte assassination and the disbanding of Lehi, he joined the IDF and was placed in the Artillery Corps. He was discharged at the rank of captain.
Afterwards, he completed his education and studied to be an accountant. He was part of the Herut Movement from its first days, as a central committee members and activist. Eventually, he turned to municipal politics, and he won a seat on the Jerusalem City Council from 1956 until 1981; and from 1969 to 1979, he was acting and deputy mayor. He was also on the board of the government housing and development company.
In 1984, he was elected to the Knesset as a Likud member. Since then, he has served in many senior movement positions representing the Likud. He served on the Knesset finance committee, the state control committee, the environment and internal affairs committee, the defense and foreign affairs committee, and the economic affairs committee. For three terms, he was the head of the Israeli delegation in the Interparliamentary Union, and for four terms he was a member of the council. In 1996, he became minister of health, serving until 1999. He remained in Knesset until 2002, when he became the head of Israel Bonds.
In 1963, Yehoshua married Rachel Pascha ibn Ezra, a tour guide, a tenth-generation resident of the Land of Israel. They have three daughters (Esther, Sigalit and Ayala) and a son (Amir).
They lived in Jerusalem for many years.