On the relation provable equivalence and on partitions in effectively inseparable sets

We generalize a well-knownSmullyan's result, by showing that any two sets of the kindCa = {x/⊢ x↔a} andCb = {x/⊢ x↔b} are effectively inseparable (if I ⌿ ↔b). Then we investigate logical and recursive consequences of this fact (see Introduction).