Is the em dash appropriate in place of a comma? [duplicate]

I am trying to learn the rules for the em dash. My colleagues and I were having an argument about this. One wrote this sentence for his students

We have just derived a fundamental trigonometric identity—the Pythagorean identity.

I suggested that the em dash cannot properly take the place of a comma. My colleague, however, insisted this was proper punctuation usage for emphasis. I have my doubts but I am not an expert. I would appreciate any input on this.

Your colleague is correct.

Here is what the Chicago Manual of Style says about it:

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6.85: Em dashes instead of commas, parentheses, or colons

The em dash, often simply called the dash, is the most commonly used and most versatile of the dashes. (In British usage, spaced en dashes are used in place of em dashes; see 6.83.) Em dashes are used to set off an amplifying or explanatory element and in that sense can function as an alternative to parentheses (second and third examples), commas (fourth and fifth examples), or a colon (first example)—especially when an abrupt break in thought is called for.

It was a revival of the most potent image in modern democracy—the revolutionary idea.
The influence of three impressionists—Monet, Sisley, and Degas—is obvious in her work.
The chancellor—he had been awake half the night—came down in an angry mood.
She outlined the strategy—a strategy that would, she hoped, secure the peace.
My friends—that is, my former friends—ganged up on me.

To avoid confusion, the em dash should never be used within or immediately following another element set off by an em dash (or pair of em dashes). Use parentheses or commas instead.

The Whipplesworth conference—which had already been interrupted by three demonstrations (the last bordering on violence)—was adjourned promptly.

or

The Whipplesworth conference—which had already been interrupted by three demonstrations, the last bordering on violence—was adjourned promptly.

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