Does the declaration declare more than just independence?

In Derrida’s “Declarations of Independence,” he discusses on what basis the signature of the DOI becomes a legitimate manner to both express and in this expression, enact independence. I think his process of declaration can also be applied to the rights stated in the DOI, which Armitage implies in “The Declaration of Independence and International Law,” had yet to be formally recognized by an established government even though they had little bearing on the act of gaining independence.
Derrida addresses the rights stated in the DOI and argues that it is one reason which compelled the signers to declare independence. He writes, “… they should declare the causes which impel them to separation: ‘We hold these truths to be self-evident: that all men are created equal; that they are endowed by their creator with inalienable Rights …(Derrida 11),” specifically of “Life, Liberty and the pursuit of Happiness (DOI),” which are infringed upon. While these may be “self-evident,” it wasn’t to Britain at the time, aligning these rights alongside independence; although the states “are” already independent, it is not till the declaration of this independence that it becomes true – the same can be said about these rights. The signature grants itself recognition of these rights which it intrinsically has but is not recognized, and furthermore is more legitimate in its justification to sign because it has them and recognizes them by creating a state that does so upon signing.
However, as Armitage points out, this assertion was “strictly subordinate (44),” as one of many justifications for separation, leading to the question its presence. Because these rights are new in terms of their formal recognition (distinct from and not yet, legally protected) it can be assumed as Armitage asserts that their presence is for future use. For a government looking to eventually protect these “self-evident rights” these rights must be evident since the moment of formation, making their appearance important in the DOI. Given this, the signers had to call on a higher power than the British Crown to grant these rights legitimacy in the declaration, stating “the Creator” has endowed them (DOI). This section of the DOI therefore links these rights with God as a manner to unify and appeal to common beliefs, although this intertwinement may later seem contradictory when granting these rights legal protection in what will be a secular state.