What's the difference between $\equiv$ and $\leftrightarrow$ in a formal proof?
Solution 1:
In propositional calculus there is a connective $\rightarrow$ and often one for $\leftrightarrow$, which is a part of the formal language and how we create new propositions; and there is another form, $\implies$ and $\iff$ (also written as $\cong$) which are meta-statements.
Whereas $\varphi\leftrightarrow\psi$ is a proposition, $\varphi\iff\psi$ is a statement about propositions.
It is true, however, that $\varphi\iff\psi$ is true, if and only if $\varphi\leftrightarrow\psi$ is a tautology. However there is a difference between a proposition, and a statement about propositions; and I cannot stress this difference enough in this post.
Solution 2:
With respect to the "$\equiv$" symbol (my comment was getting long!):
This is more in the way of speculation, but it seems that "$a \equiv b$" is used in some (not all) contexts to denote "$a$ is identically $b$", or in other contexts to convey that $a$ and $b$ are essentially equivalent (with respect to some equivalence-relation, e.g. congruence modulo $n$, or geometric congruence, or truth-functionality, or... etc.), again, depending on the contexts in which it's being used.
This would be consistent with the frequent use of the symbol "$\equiv$" in introductory logic texts to convey that $a \equiv b$ holds (is true) whenever $a$ and $b$ evaluate to the same truth value. It's also consistent with what you've read: "$\equiv$" can be read as "can be replaced in a logical proof with...", in the sense that if "$p\equiv q$", then replacing every occurrence of $p$ with $q$ (or vice-versa) will not change the truth-value of any propositions thus impacted.
I know that I've used "$\equiv$" and "$\leftrightarrow$" interchangeably on this site, when answering, e.g., questions about propositional logic: partly for reasons related to trying to match the notation used in the question, and partly due to being careless and/or not necessarily knowing better.
Solution 3:
Some author as do use $\equiv$ to mean "if and only if". Another option, which I use from time to time, is to use $\equiv$ as a sort of equality symbol when I am talking about formulas. If I want to give the name $\phi$ to the formula "$a=b"$, I don't want to write $$ \phi = a = b $$ so I write $$ \phi \equiv a = b $$ instead. This allows me to avoid using quotation symbols.
This latter usage is not documented anywhere, I as far as I know. I am not aware of any book that describes it in detail. But I can explain my motivations for using it:
I don't want to re-use the $=$ sign for another purpose; the first displayed formula is not appealing. But quotation marks in displayed formulas are also unappealing.
When I write $\phi \equiv a = b$, I sometimes don't care if $\phi$ is the formula $a=b$ or is just equivalent to it. But for the rest of the proof I will act as if it is that formula. So the $\equiv$ symbol, which connotes "equivalent", suggests that all that matters in the beginning is that $\phi$ is equivalent to $a=b$.