Where in Shas can I find the statement that with every chumra one takes on, he is in essence going to be taking a on a kula somewhere else? I'm pretty sure it's in Beitza somewhere...
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The gemara does not say that every stringency leads to a leniency, but sometimes it does, and thus the gemara may challenge that a certain stringency is a חומרא דאתי לידי קולא - a stringency which leads to a leniency, for example in Pesachim 48b, Yevamos 30b, Bava Kama 11a and Niddah 24b.
The closest thing I could find is this, on 16a: