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Granted, Great Britain and France came in to defend their assets, however Israel had a different motivation, one which is critical to know and realize in order to fully comprehend the war in 1956. 

It has to do with Egyptian actions which predated the 1956 war:

Egypt had maintained its state of belligerency with Israel after the 1949 armistice agreement was signed. 

The first manifestation of this was the closing of the Suez Canal to Israeli shipping.  (Egypt would repeat this step in 1967, before the June war)

On August 9, 1949, the UN Mixed Armistice Commission upheld Israel’s complaint that Egypt was illegally blocking the canal.