The sport you are referring to is called Gaelic football, which is one of Ireland's national sports. There are a few things in it which you would find in Rugby or in soccer, but it is a totally unique game in its own right with its own rules and history. So it would be wrong to say it is rugby and football together. Many ball sports have things in common with other ball sports, but they are unique. That is true of Gaelic Football.
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No its not. The only similarity is that you can kick the ball.
ANSWER
No. Soccer and rugby actually evolved from the same game, and American football in turn evolved out of rugby.
Rugby and soccer split apart from each other in the mid-19th century, with clubs that wanted to use the hands in open play developing the rugby game, and those that didn't going off to form association football, or soccer. (There were other differences, but use of the hands was one of the big ones.)
American colleges later standardized the rules for their football clubs based on the English rugby code, and between the 1870s and the early 1900s, that game took on its own flavor, including a down-and-distance system, a line of scrimmage replacing the scrum, and the legalization of the forward pass.
If you watch a rugby match, you're pretty much getting a glimpse of what American football looked like 100 years ago. But there's really no similarity to soccer at all.
Gaelic Games
The main sport on a national level in Ireland is the national indigenous GAA games of hurling and Gaelic football which enjoy nationwide popularity ahead of rugby and soccer.