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In other words, $P(x)\iff Q(x)$ implies that $\forall x \in D, P(x) \leftrightarrow Q(x)$ where $D$ is the relevant domain. Or, that "$\iff$" applies when talking about logic while "$\leftrightarrow$" is used in a formula.

But to address the question as to whether $\iff$ can serve as a replacement for $=$, I think that translating it into English is useful. Replace "$\iff$" with "is necessary and sufficient for" and say "$=$" as "equals" (the same goes for "$\equiv$" and "logically equivalent to") and I think that you'll see that they are different, conceptually and syntacticly, only in certain circumstances, or "when doing logic, we speak logic" (I often use them when taking notes on non-math topics, and my brain knows what I mean regardless of symbol choice :) ).

I don't know if this answers the question, but I hope it was useful, (and others, feel free to correct any misconceptions I may have.)

1 [made Community Wiki]

In other words, $P(x)\iff Q(x)$ implies that $\forall x \in D, P(x) \leftrightarrow Q(x)$ where $D$ is the relevant domain. Or, that "$\iff$" applies when talking about logic while "$\leftrightarrow$" is used in a formula.

But to address the question as to whether $\iff$ can serve as a replacement for $=$, I think that translating it into English is useful. Replace "$\iff$" with "is necessary and sufficient for" and say "$=$" as "equals" (the same goes for "$\equiv$" and "logically equivalent to") and I think that you'll see that they are different, conceptually and syntacticly, only in certain circumstances, or "when doing logic, we speak logic" (I often use them when taking notes on non-math topics, and my brain knows what I mean regardless of symbol choice :) ).

I don't know if this answers the question, but I hope it was useful, (and others, feel free to correct any misconceptions I may have.)