Nah, history doesn't fully back you up there....Israel decided to face off with Egypt after Egypt decided to stop allowing ships in. Syria then decided to try and get in on this action all on their own.
In May–June 1967, in preparation for conflict, the Israeli government planned to confine the confrontation to the Egyptian front, whilst taking into account the possibility of some fighting on the Syrian front.
Syrian front 5–8 June
Syria largely stayed out of the conflict for the first four days.
False Egyptian reports of a crushing victory against the Israeli army and forecasts that Egyptian forces would soon be attacking Tel Aviv influenced Syria's decision to enter the war
In the months prior to the outbreak of the Six-Day War in June 1967, tensions again became dangerously heightened: Israel reiterated its post-1956 position that another Egyptian closure of the Straits of Tiran to Israeli shipping would be a definite casus belli. In May 1967, Egyptian president Gamal Abdel Nasser announced that the Straits of Tiran would again be closed to Israeli vessels. He subsequently mobilized the Egyptian military into defensive lines along the border with Israel and ordered the immediate withdrawal of all UNEF personnel.
On 5 June 1967, as the UNEF was in the process of leaving the zone, Israel launched a series of airstrikes against Egyptian airfields and other facilities in what is known as Operation Focus. Egyptian forces were caught by surprise, and nearly all of Egypt's military aerial assets were destroyed, giving Israel air supremacy. Simultaneously, the Israeli military launched a ground offensive into Egypt's Sinai Peninsula as well as the Egyptian-occupied Gaza Strip. After some initial resistance, Nasser ordered an evacuation of the Sinai Peninsula; by the sixth day of the conflict, Israel had occupied the entire Sinai Peninsula. Jordan, which had entered into a defense pact with Egypt just a week before the war began, did not take on an all-out offensive role against Israel, but launched attacks against Israeli forces to slow Israel's advance. On the fifth day, Syria joined the war by shelling Israeli positions in the north.
In May–June 1967, in preparation for conflict, the Israeli government planned to confine the confrontation to the Egyptian front, whilst taking into account the possibility of some fighting on the Syrian front. Syrian front 5–8 June
Syria largely stayed out of the conflict for the first four days.
False Egyptian reports of a crushing victory against the Israeli army and forecasts that Egyptian forces would soon be attacking Tel Aviv influenced Syria's decision to enter the war
https://en.wikipedia.org/wiki/Six-Day_War#Golan_Heights
Two thirds of the area was depopulated and occupied by Israel following the 1967 Six-Day War and then effectively annexed in 1981
https://en.wikipedia.org/wiki/Golan_Heights
In the months prior to the outbreak of the Six-Day War in June 1967, tensions again became dangerously heightened: Israel reiterated its post-1956 position that another Egyptian closure of the Straits of Tiran to Israeli shipping would be a definite casus belli. In May 1967, Egyptian president Gamal Abdel Nasser announced that the Straits of Tiran would again be closed to Israeli vessels. He subsequently mobilized the Egyptian military into defensive lines along the border with Israel and ordered the immediate withdrawal of all UNEF personnel.
On 5 June 1967, as the UNEF was in the process of leaving the zone, Israel launched a series of airstrikes against Egyptian airfields and other facilities in what is known as Operation Focus. Egyptian forces were caught by surprise, and nearly all of Egypt's military aerial assets were destroyed, giving Israel air supremacy. Simultaneously, the Israeli military launched a ground offensive into Egypt's Sinai Peninsula as well as the Egyptian-occupied Gaza Strip. After some initial resistance, Nasser ordered an evacuation of the Sinai Peninsula; by the sixth day of the conflict, Israel had occupied the entire Sinai Peninsula. Jordan, which had entered into a defense pact with Egypt just a week before the war began, did not take on an all-out offensive role against Israel, but launched attacks against Israeli forces to slow Israel's advance. On the fifth day, Syria joined the war by shelling Israeli positions in the north.
https://en.wikipedia.org/wiki/Six-Day_War