If someone by mistake didn't say atah y'ztarteh (the special tefillah we say by musaf on Shabbos that is also rosh chodesh) and instead davened a regular musaf on shabbos rosh chdoesh, what's the din b'dieved?
I once heard that if someone davened a normal musaf on SHabbos Chol HaMoed Pesach they would be yotzee b'dieved since in musaf we say "musafim c'hilchasem" in a normal musaf of Shabbos it implies all musafim, even that of the yom tov. Perhaps so to here, even that of Rosh Chodesh.
Is this true? What's the din b'dieved in the original question? Who speaks about this? What other proofs could there be one way or another?