Biconditional in terms of NAND

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$p \iff q \dashv \vdash \paren {\paren {p \uparrow p} \uparrow \paren {q \uparrow q} } \uparrow \paren {p \uparrow q}$

where $\iff$ denotes logical biconditional and $\uparrow$ denotes logical NAND.


\(\ds p \iff q\) \(\dashv \vdash\) \(\ds \neg \paren {p \oplus q}\) Exclusive Or is Negation of Biconditional
\(\ds \) \(\dashv \vdash\) \(\ds \neg \paren {\paren {p \lor q} \land \neg \paren {p \land q} }\) Definition of Exclusive Or
\(\ds \) \(\dashv \vdash\) \(\ds \neg \paren {\paren {p \lor q} \land \paren {p \uparrow q} }\) Definition of Logical NAND
\(\ds \) \(\dashv \vdash\) \(\ds \neg \paren {\paren {\paren {p \uparrow p} \uparrow \paren {q \uparrow q} } \land \paren {p \uparrow q} }\) Disjunction in terms of NAND
\(\ds \) \(\dashv \vdash\) \(\ds \paren {\paren {p \uparrow p} \uparrow \paren {q \uparrow q} } \uparrow \paren {p \uparrow q}\) Definition of Logical NAND