I proved a theorem I heard from another researcher. Can I publish my results?
If the proof is yours then of course you can publish it. And it is your duty to do it, because the literature is incomplete without it. 

Basically you say (but in more formal language):


at the Holcombe Colloquium (August 2015), R.J. Blenkinsop asserted the following, without giving a proof: insert theorem here.
no proof was given at Holcombe and there appears to be none in the literature. 
this theorem is interesting and useful, for instance in the context of… insert description of &amp; reference to your paper
here is a proof


You have thus acknowledged Blenkinsop as the source of the idea and asserted the originality of your own contribution. Both halves of this action are true and ethically sound. 

Of course a referee may contact Blenkinsop who, now that fame and glory are involved, may take the trouble to look up his own proof. But unless it's already published ("see my Simple Sums for Simple Minds, page 2"), you still have priority.