Conditional in terms of NAND

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

where $\implies$ denotes the conditional and $\uparrow$ denotes the logical NAND.

Proof 1

\(\ds p \implies q\) \(\dashv \vdash\) \(\ds \neg \paren {p \land \neg q}\) Conditional is Equivalent to Negation of Conjunction with Negative
\(\ds \) \(\dashv \vdash\) \(\ds p \uparrow \neg q\) Definition of Logical NAND
\(\ds \) \(\dashv \vdash\) \(\ds p \uparrow \paren {q \uparrow q}\) NAND with Equal Arguments


Proof 2

\(\ds p \implies q\) \(\dashv \vdash\) \(\ds \neg p \lor q\) Rule of Material Implication
\(\ds \) \(\dashv \vdash\) \(\ds \neg p \lor \neg \neg q\) Double Negation Introduction
\(\ds \) \(\dashv \vdash\) \(\ds p \uparrow \neg q\) NAND as Disjunction of Negations
\(\ds \) \(\dashv \vdash\) \(\ds p \uparrow \left({q \uparrow q}\right)\) NAND with Equal Arguments