Validity of Argument (Part 2)

∀A,B,C,K_1,K_2∈N. By this I mean they are all elements of the natural numbers. 1. ∃K_1 (K_1≤1) 2. ∃A,B,C,K_2 (A^(K_2 )+B^(K_2 )=C^(K_2 ) ) 3. ∃K_1 (K_1≤1)∧∃A,B,C,K_2 (A^(K_2 )+B^(K_2 )=C^(K_2 ) ) 4. ∃A,B,C,K_2 (A^(K_2 )+B^(K_2 )=C^(K_2 ) )∧∃K_1 (K_1≤1) 5. ∃A,B,C,K_2 (A^(K_2 )+B^(K_2 )=C^(K_2 ) )→∃K_1 (K_1≤1) 6. ∀K_1 (K>1)→∀A,B,C,K_2 (A^(K_2 )+B^(K_2 )≠C^(K_2 ) ) Premise 1 is true because there does exist a natural number less than or equal to 1. Premise 2 is true because there does exist A,B,C, and K such that A^K+B^K=C^K. Premise 3 is true because of both Premises 1 and 2. Premise 4 is true because of Premises 1, 2, and 3. Premise 5 is true because it is the contrapositive of Premise 4. Now with regard to Premise 5, if I let K_1=K_2=2, the antecedent is true but the consequent is false, making Premise 5 false. Yet how can Premise 5 be false if all the premises before it are true?