Fermat's Last Theorem and the Principle of Duality

Let A,B,C,K be natural numbers. Let T signify true. Let F signify false. So we have ∀K(K≤2→∃A,B,C(A^K+B^K=C^K ))=T. But this is equivalent to ∀K(K>2∨∃A,B,C(A^K+B^K=C^K ))=T. By the principle of duality we then have ∀K(K>2∧∃A,B,C(A^K+B^K=C^K ))=F . Also consider the following. Again, let A,B,C,K be natural numbers. ∃K(K≤2∧∃A,B,C(A^K+B^K=C^K ))=T. By the principle of duality we then have ∃K(K≤2∨∃A,B,C(A^K+B^K=C^K ))=F. But this is equivalent to ∃K(K>2→∃A,B,C(A^K+B^K=C^K ))=F