Israel was technically still at war with Egypt. '48 ended with an armistice, not a peace. They viewed Nasser as fundamentally dangerous, and his seizure of the Suez Canal as both an example of his dangerousness as well as being to his advantage.

why did Israel invade first without military support from the European countries?

Israel did have that support. In fact they got it in writing which is how we know that Anthony Eden (among others) was lying about it. The point of the plan was Israel would re-initiate hostilities and then France and Britain would swoop in and "peace keep" their way into re-securing canal. In other words it was perhaps the most sophisticated, extensive and brazen examples of a "false flag" operation in world history.