Do you still believe that theirs is a song of ice and fire if Jon Snow is the ice and fire himself?

At this point, we’re inclined to believe that it’s a combination of Jon and Daenerys, not just Jon–so yes, we believe that theirs is a song of ice and fire.

“He has a song,” the man replied. “He is the prince that was promised, and his is the song of ice and fire.” He looked up when he said it and his eyes met Dany’s, and it seemed as if he saw her standing there beyond the door. “There must be one more,” he said, though whether he was speaking to her or the woman in the bed she could not say. “The dragon has three heads.” He went to the window seat, picked up a harp, and ran his fingers lightly over its silvery strings. Sweet sadness filled the room as man and wife and babe faded like the morning mist, only the music lingering behind to speed her on her way.  (Daenerys IV, ACOK)

Dany’s vision in the House of the Undying is of Rhaegar, ruminating on the prophecy he had discovered as a boy and which he thought meant he was the Prince that was Promised and then, subsequently, that his sone Aegon was the Prince that was Promised.  He believes that the song of Ice and Fire is that of the Prince that was Promised.

The problem is that Martin very much plays with how prophecy works:

“An ant who hears the words of a king may not comprehend what he is saying,” Melisandre said, “and all men are ants before the fiery face of god. If sometimes I have mistaken a warning for a prophecy or a prophecy for a warning, the fault lies in the reader, not the book. But this I know for a certainty—envoys and pardons will not serve you now, no more than leeches. You must show the realm a sign. A sign that proves your power!” (Davos V, ASOS)

“No,” the old man said. “It must be you. Tell them. The prophecy…my brother’s dream…Lady Melisandre has misread the signs. Stannis…Stannis has some of the dragon blood in him, yes. His brothers did as well. Rhaelle, Egg’s little girl, she was how they came by it…their father’s mother…she used to call me Uncle Maester when she was a little girl. I remembered that, so I allowed myself to hope…perhaps I wanted to…we all deceive ourselves, when we want to believe. Melisandre most of all, I think. The sword is wrong, she has to know that…light without heat…an empty glamor…the sword is wrong, and the false light can only lead us deeper into darkness, Sam. Daenerys is our hope. Tell them that, at the Citadel. Make them listen. They must send her a maester. Daenerys must be counseled, taught, protected. For all these years I’ve lingered, waiting, watching, and now that the day has dawned I am too old. I am dying, Sam.”  (Samwell IV, AFFC)

“Born amidst salt and smoke, beneath a bleeding star. I know the prophecy.” Marwyn turned his head and spat a gob of red phlegm onto the floor. “Not that I would trust it. Gorghan of Old Ghis once wrote that a prophecy is like a treacherous woman. She takes your member in her mouth, and you moan with the pleasure of it and think, how sweet, how fine, how good this is…and then her teeth snap shut and your moans turn to screams. That is the nature of prophecy, said Gorghan. Prophecy will bite your prick off every time.” (Samwell V, AFFC)

So Martin’s clearly not setting up a prophecy to be at face value, which means that Rhaegar’s view of the Song of Ice and Fire is not so much about the true meaning of the prophecy, so much as Rhaegar’s failure in interpreting it accurately.

So yes, you can say that Jon’s is the Song of Ice and Fire, that he is of both Ice (Stark) and Fire (Targaryen), that he was born amidst salt and smoke (as people argue).  Or you can keep your mind open to the idea that it might not be just Jon, especially given that almost all of that prophetic fulfillment also has possibilities in Daenerys and, of course, 

“No one ever looked for a girl,” he said. “It was a prince that was promised, not a princess. Rhaegar, I thought… the smoke was from the fire that devoured Summerhall on the day of his birth, the salt from the tears shed for those who died. He shared my belief when he was young, but later he became persuaded that it was his own son who fulfilled the prophecy, for a comet had been seen above King’s Landing on the night Aegon was conceived, and Rhaegar was certain the bleeding star had to be a comet. What fools we were, who thought ourselves so wise! The error crept in from the translation. Dragons are neither male nor female, Barth saw the truth of that, but now one and now the other, as changeable as flame. The language misled us all for a thousand years. Daenerys is the one, born amidst salt and smoke. The dragons prove it.” Just talking of her seemed to make him stronger. “I must go to her. I must. Would that I was even ten years younger.” (Samwell IV, AFFC)