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What's the consistency status/strength of this limitation principle?

This principle is inconsistent: consider the formula $\theta(x)$ = "$x^+$ is the smallest infinite cardinal at which $\mathsf{CH}$ fails." The formula $\theta$ cannot hold on more than one ...
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Which extension of ZFC proves that ZFC can only prove CH satisfied by the first two sets?

The statement you propose is equivalent to consistency of GCH failing for all (infinite) cardinals. This problem is well-studied. Of course the theory ZFC+Con(ZFC+"GCH fails everywhere") ...
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