In season 5, episode 10, the Atlantis Stargate is destroyed. At the beginning of episode 11 it is shown as big hole where it once existed.
In later episodes they use the Atlantis Stargate. How is this possible?
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At the end of the following episode is some conversation you must have missed.
From 5x11, "The Lost Tribe":
Jackson: "I don't think I'll be around to find out, though."
McKay: "Yeah I heard they're, um, sending you back to Earth."
Jackson: "Yeah, I guess I need some major patching up and recovery, and Landry prefers I do that there instead of here, so..."
McKay: "Well, the Daedalus has already picked up a Gate, uh, leftover from the old Gate bridge, and uh, my team is installing it as we speak, so we should have you home in, uh, no time at all."
Jackson: "That's good, that is... good."
It's fairly well-established that Stargates are movable/interchangeable.
In this specific instance, one of the Carter-McKay bridge gates was used to replace the one destroyed by the Attero device. These gates had become redundant after the destruction of the midway station.
From the Stargate wikia
The Stargate was eventually replaced with one of the gates from the McKay/Carter Intergalactic Gate Bridge. The gate shield still operates with the new gate, suggesting that the shield generator for the Stargate shield is either not actually inside the Stargate itself or it is somehow an extension of Atlantis' primary shield generators. Also, because dialing another galaxy is also not dependent upon the actual gate, that function remains as well.
THe stargate was replaced, just like the Earth stargate did. The order from the Earth stargate goes like this: