Was Voldemort really searching for the Hallows

Priori Incantatem is not simply connecting beams of spells. It is rare magic that occurs when brother wands are forced to battle, while it does result in the charms fired connecting, it also results in a dome of light surrounding the two duelling wizards (in the books it also moves them away from the crowd of Deatheaters). There is a point that the spells meet, and when this is pushed back into either wand, it will force echoes of that wand's previous spells to appear. This only happens once in the entire series of 8 movies: When Voldemort duels Harry in the graveyard in Harry Potter and the Goblet of Fire. Remember that Harry's wand with the core of the Phoenix feather was identical to Voldemort's, as mentioned in the first book and movie, the wand chooses the wizard, and because it turns out that Harry had a fragment of Voldemort's soul, they ended up with "twin wands". (Voldemort used his wand to kill Cedric Diggory, Frank Bryce (the Muggle caretaker of the Riddle house), and James and Lily Potter and echoes of these people come out of Voldemort's wand and help Harry escape.) What has confused people, is that in Harry Potter and the Order of the Phoenix, an artistic decision was made to have the Dumbledore & Voldemort duel appear more visually impressive and take advantage of the scene being shot in IMAX by having the lava-like connection of spells (such a connection does not occur in the books). This was not priori incantatem. But the effect has remained in the following movies. The key difference being that there is no dome of light, and no echoes being forced from wands due to the meeting point of spells being forced into one of the wands. The only instance of a spell being forced into a wand is in the instance of the start of Harry Potter and the Deathly Hallows: Part 1 when Voldemort uses Lucius Malfoy's wand, and the wand is destroyed. This was not Priori Incantatem either. Harry does not cast a spell, the wand acts of its own accord (Harry is prone in pain), and destroys Lucius's wand. In the final duel, it is not priori incantatem either, the Elder Wand is refusing to kill its master, Harry, and the avada kedavra curse backfires on Voldemort. Shortly after the novel-release of Harry Potter and the Deathly Hallows, J.K Rowling revealed that the core of the Elder Wand was a Thestral tail hair, and the Elder Wand was the only wand in the world that had that kind of core, and because of that the Elder Wand can't create Priori Incantatem with any other wand, because there is no "twin wand". Edit