$\equiv$ 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