"Justice has been done," Khan said. "It means this fight (against Garcia) will show who is the best in the division."
Khan appealed following the points defeat against Peterson, with the WBA saying there were
"multiple irregularities" in the fight.
The IBF is still to make a decision as to whether to reverse the original fight result and return its belt to Khan.
The Bolton-born boxer said: "I'm glad the WBA are reinstating me as champion; it means I walk into this fight as world champion.
"There is a chance to win the WBC belt as well but, not only that, the Ring magazine title will be on the line. There are five main belts in this division and I will have won four of them if I win this fight.
"The only one I have not won is the WBO one, but I tried to fight Tim Bradley for that and he turned me down. If I win against Garcia, it means I have done everything I want to do in this division."
With the WBA title now on the line, the light-welterweight title fight against Garcia gains added significance.
Khan claims his contest easily eclipses David Haye's heavyweight clash with Dereck Chisora in London on the same night.
The Haye-Chisora bout has attracted a lot of interest following their much-publicised brawl in Munich in February.
"Boxing fans know my fight is bigger," Khan told BBC Radio 5 live Breakfast. "There's a world title on the line."
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