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