What's the difference between these two theorems? On p. 124 of Gamut's Logic, Language and Meaning, v. 1, there's the following theorem:

Theorem 5
If $\phi$ and $\psi$ are equivalent, $\phi$ is a subformula of $\chi$, and $[\psi/\phi]\chi$ is the formula obtaind by replacing this subformula $\phi$ in $\chi$ by $\psi$, then $\chi$ and $[\psi/\phi]\chi$ are equivalent.

Then, after the proof-sketch of this theorem, there's the following (my emphasis):

The same reasoning also proves the following, stronger theorem (in which $\phi$, $\psi$, $\chi$, $[\psi/\phi]\chi$ are the same as above):
Theorem 6 (Principle of extensionality for sentences in predicate logic)
$\phi \leftrightarrow \psi \vDash \chi \leftrightarrow [\psi/\phi]\chi$.

I have a hard time telling these two apart, let alone discerning that 6 is stronger than 5.
I read the expression to the left of the $\vDash$ in 6 as "$\phi$ and $\psi$ are equivalent", which is exactly the premise of 5.  Likewise, I read the expression to the right of the $\vDash$ in 6 as "$\chi$ and $[\psi/\phi]\chi$ are equivalent", exactly the conclusion on 5.
The only possible difference I can guess is that maybe the equivalence in 5 is "logical equivalence", and the one in 6 is "material equivalence".  If so, I still don't understand why 6 is stronger.  Even if its premise is weaker, so is its conclusion.

UPDATE
OK, here's one possible elucidation.  Let's suppose that, in 5, "equivalent" means "logically equivalent".  This means that, for all models, $\phi \leftrightarrow \psi$ is true.  So 5 could be interpreted as a corollary of 6.  In words,

If $\phi \leftrightarrow \psi$ is true for all models, then $\chi \leftrightarrow [\psi/\phi]\chi$ is true for all models.

The reason why 6 is stronger than 5 is that 5 is a statement about the case where $\phi \leftrightarrow \psi$ is true for all models, whereas 6 does not have this requirement.  In fact, 6 that for all the models for which $\phi \leftrightarrow \psi$ is true (however many these may be), $\chi \leftrightarrow [\psi/\phi]\chi$ is also true.
Is this the correct interpretation?
 A: No, you don't have the correct interpretation.  
The $\phi \leftrightarrow \psi$ in 6 is a logic statement that contains a material conditional ... which we typically read as "$\phi$ if and only if $\psi$" ...  but that is not the same as saying that $\phi$ and $\psi$ are logically equivalent. And when logicians talk about 'equivalence', they  mean 'logical equivalence', so that's the same thing. So no, don't interpret a biconditional as a statement about (logical) equivalence!
Example:
$\phi$ is (logically) equivalent to $\neg \neg \phi$ ... but that we typically write as $\phi \Leftrightarrow \neg \neg \phi$ .  Note the $\Leftrightarrow$ is the (meta!)-logical symbol we use to talk about equivalence .. again, this is not the same as $\leftrightarrow$
On the other hand, in logic we may encounter something like $p \leftrightarrow q$ ... but clearly $p$ and $q$ are not logically equivalent.
5 talks about the substitution of logically equivalent statements, i.e. where $\phi \Leftrightarrow \psi$.  
But 6 talks about a substitution principle involving a biconditional $\phi \leftrightarrow \psi$. 
5 is a special case of 6 (and thus 6 is stronger), since if $\phi \Leftrightarrow \psi$, then obviously it will be true that $\phi \leftrightarrow \psi$, but not vice versa.
