Dtyjlui

I am aware that there are many previously asked questions regarding the proof that the duals of two equivalent compound propositions containing only logical operators ∨, ∧, and ¬ are also equivalent, although I have not yet been content with the solutions I have found.

If possible, I would like someone qualified to see if I am on the right track with my current proof of the above. Please note that I am only at a relatively basic level for proof-writing, although I wish to advance this whenever possible.

Below is a definition of the dual of a compound proposition containing only logical operators ∨, ∧, and ¬, found in Discrete Mathematics and Its Applications, 7th Edition, by Rosen, followed by the relevant theorem and current draft of my proof.

The dual of a compound proposition that contains only the
logical operators ∨, ∧, and ¬ is the compound proposition
obtained by replacing each ∨ by ∧, each ∧ by ∨, each T
by F, and each F by T. The dual of s is denoted by s∗.

Theorem:

The duals of two equivalent compound propositions are also equivalent.

Proof:

Let $s_0$, $s_n$ be two arbitrary compound propositions containing
only logical operators ∨, ∧, or ¬, such that $s_0$ $equiv$ $s_n$
after transforming $s_0$ with n logical equivalences containing
only logical operators ∨, ∧, or ¬.

Let $t_0$, $t_1$, $ldots$ , t_n-1 be an ordered sequence of
n transformations using logical equivalences only containing the ∨, ∧,
or ¬ operators such that $t_i$($s_i$) = s_i+1, for 0 $leq$
i $leq$ n-1, where $s_i$, s_i+1 are compound propositions, and let
$s_0^*$, $s_1^*$, $ldots$ , $s_n^*$ be the
corresponding duals for $s_0$, $s_1$, $ldots$ , $s_n$, respectively.

We wish to find an ordered list of m transformations, $t_0^*$, $t_1^*$,
$ldots$ , t_m-1$^*$ such that $t_i^*$($s_i^*$) =
s_i+1$^*$,
for 0 $leq$ i $leq$ m-1, where $s_i$,
s_i+1 are compound propositions.

Since every logical equivalence for a compound proposition containing only
the ∨, ∧, or ¬ operators has a dual equivalent, that is, a complementary
logical equivalence for the dual of the compound proposition, we let the
logical equivalence transformation $t_i^*$ be the dual equivalent of the
transformation $t_i$ when the transformation is performed on a compound
proposition, and otherwise the same transformation when performed on an
individual element,
for 0 $leq$ i $leq$ n-1.

Thus, $t_i^*$($s_i^*$) = s_i+1$^*$, for 0 $leq$ i $leq$ n-1,
is a valid ordered sequence of logical equivalence transformations, and $s_0^*$
$equiv$ $s_n^*$. Since $s_0^*$ and $s_n^*$ were chosen to be arbitrary,
true for all such.

End of proof.

I feel that the first statement in last paragraph of my proof is not explicit enough, although I am not currently able to think of what is additionally needed.

Please help if possible. Thank you!