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So, if you start with air at 95 degrees with relative humidity of 70% and cool that same air to 75 degrees the relative humidity will increase.
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That's not true if you use an air conditioner to cool the air because the air "conditioning" process removes moisture as it cools. That's what the condensate drain line is for... to carry away the moisture that has been condensed out of the air. This leaves you with cooler, drier air for your living space.
Now if you were living in a cave and had a giant fan to pull hot air into your cave where it was much cooler, then the humidity of your cave would be higher than the outside air, but that's not what happens with air conditioning in our homes because air conditioners REMOVE moisture. That's why they're called air CONDITIONERS and not air COOLERS... because they "condition" the air, i.e. remove moisture AND cool the air.