What's the difference between these two theorems?

Question

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.

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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?

Answer 1

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.