\( \equiv \) Theorem 2: p \( \equiv \) p

\( \hspace{2 cm}\) p \( \equiv \) p
\( \hspace{0.5 cm} = \) the \( \equiv \) Identity Axiom. \( p \equiv p \equiv true \)
\( \hspace{2 cm}\) true. QED

\( \equiv \) Theorem 1: true

\( \hspace{2 cm} true \equiv p \equiv p \), Axiom, Identity of \( \equiv \),
\( \hspace{0.5 cm} = \) Leibniz, E = Identity Axiom above, \(p = p \equiv p, q = true \)
\( \hspace{2 cm} true \equiv true \)
\( \hspace{0.5 cm} \equiv \) Axiom, Identity of \( \equiv \),\(true \equiv p \equiv p \),
\( \hspace{2 cm} \) true

EQ Axioms and Advice

(1) Axiom: Associativity of \(\equiv\): p \(\equiv\) (q \(\equiv\) r) \( \hspace{0.2 cm} \equiv \hspace{0.2 cm} \) (p \(\equiv\) q ) \(\equiv\) r

(2) Axiom: Symmetry of \(\equiv\): p \(\equiv\) q \( \hspace{0.2 cm} \equiv \hspace{0.2 cm} \) q \(\equiv\) p

(3) Axiom: Identity of \(\equiv\): true \(\equiv\) q \(\equiv \) q