Is this sequence legal?

[MiamiWadeCounty]

I see this sometimes during halfcourt games , where a player would track down a defensive rebound and then clear the ball out past the 3pt line by tapping/moving/pushing the ball with his chest so he can set himself with the tripple threat position once the ball has been cleared out.

So the sequence goes like this:

1. Player pushes def rebound past out the 3pt line with chest (or sometimes with a headbutt).
2. Once ball is past 3pt line, player grabs ball with hands.
3. Player starts dribbling with hand.

I believe this should be a double dribble because player begins to dribble when he taps/moves/pushes ball to the floor with chest and then grabs ball with hands and dribbles again.