EQ 14 Negation, Inequivalence, False

(1) Axiom: Definition of false: false \( \hspace{0.2 cm} \equiv \hspace{0.2 cm} \) \(\neg\) true

(2) Axiom: Distributivity of \(\frac{\neg}{\equiv}\): \(\neg\) (p \(\equiv\) q) \( \hspace{0.2 cm} \equiv \hspace{0.2 cm} \) \(\neg\)p \(\equiv\) q

(3) Axiom: Definition \(\not\equiv\): (p \(\not\equiv\) q) \( \hspace{0.2 cm} \equiv \hspace{0.2 cm} \) \(\neg\)(p \(\equiv\) q)

(4) Theorem: \(\neg\)p \(\equiv\) q \( \hspace{0.2 cm} \equiv \hspace{0.2 cm} \) p \(\equiv\) \(\neg\)q

(5) Theorem: Double Negation: \(\neg\)\(\neg\)p \( \hspace{0.2 cm} \equiv \hspace{0.2 cm} \) p

(6) Theorem: Negation of false: \(\neg\)false \( \hspace{0.2 cm} \equiv \hspace{0.2 cm} \) true

(7) Theorem: (p \(\not\equiv\) q) \( \hspace{0.2 cm} \equiv \hspace{0.2 cm} \) \(\neg\)p \(\equiv\) q

(8) Theorem: \(\neg\)p\(\equiv\)p\(\equiv\)false

(9) Theorem: Symmetry of \(\not\equiv\): p \(\not\equiv\) q \( \hspace{0.2 cm} \equiv \hspace{0.2 cm} \) q \(\not\equiv\) p

(10) Theorem: Associativity of \(\not\equiv\): (p \(\not\equiv\) (q \(\not\equiv\) r)) \( \hspace{0.2 cm} \equiv \hspace{0.2 cm} \) ((p \(\not\equiv\) q ) \(\not\equiv\) r)

(11) Mutual Associativity: (p \(\not\equiv\) (q \(\equiv\) r)) \( \hspace{0.2 cm} \equiv \hspace{0.2 cm} \) ((p \(\not\equiv\) q ) \(\equiv\) r)

(12) Mutual Interchangability: p \(\not\equiv\) q \(\equiv\) r \( \hspace{0.2 cm} \equiv \hspace{0.2 cm} \) p \(\equiv\) q \(\not\equiv\) r