# Verifying a proof regarding the duals of two equivalent compound propositions also being equivalent

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 $$t_0$$, $$t_1$$, $$\ldots$$ , tn-1 be an ordered sequence of n transformations using logical equivalences only containing the ∨, ∧, or ¬ operators such that $$t_i$$($$s_i$$) = si+1, for 0 $$\leq$$ i $$\leq$$ n-1, where $$s_i$$, si+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, $$t_0^*$$, $$t_1^*$$, $$\ldots$$ , tm-1$$^*$$ such that $$t_i^*$$($$s_i^*$$) = si+1$$^*$$,
for 0 $$\leq$$ i $$\leq$$ m-1, where $$s_i$$, si+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, $$t_i^*$$($$s_i^*$$) = si+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.