Technically he is right. The strain of the air conditioning compressor, while very minor, does cause the engine to work harder to maintain the same RPM. Having said that, the difference in fuel economy will hardly be noticeable in one full tank. It would be very difficult to prove or disprove his theory because it would require driving the exact same route, at the exact same speed and acceleration for the same distance for 2 full tanks of fuel; one with the a/c on, one without. Even then, the difference would be very slight.
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