\( \{ q \to r, r \to p \} \vdash \lnot(\lnot p \land q) \)

\( \{\lnot p \land q \to r\} \vdash p \lor r \lor \lnot q\)

\( \scriptsize{\vdash ((p \land q) \to (r \lor s)) \to ((p \to r)\lor(q \to s))}\)

\( \vdash (\lnot(s \lor (p \to q))) \to p \land \lnot q \land \lnot s \)

\( \small{\{(p \to q) \land ((\lnot r \lor q) \to s) \} \vdash \lnot (p \land \lnot s) }\)

\( \small{\{(p \lor \lnot q) \to p \land r\} \vdash \lnot q \lor (\lnot p \lor r)}\)

\( \{\lnot (p \land q)\} \vdash p \to \lnot q \)

\( \{\lnot p \lor (r \to q)\} \vdash \lnot q \to \lnot (p \land r)\)

\( \{\lnot(\lnot q \land p)\} \vdash p \to q \)

\( \vdash (\lnot p \to q) \to ((p \to q)\to q)\)