Additional substitution rules show us how to remove the signal N. Remember that inference rules involve only the final letter in a string, so if N is that letter we need to see how it can be exchanged for something else.

PQAN |- PNQNO and PNQNO |- PQAN...AN subs (DeMorgan's law)
PQON |- PNQNA and PNQNA |- PQON...ON subs (DeMorgan's law)
PQCN |- PQNA and PQNA |- PQCN...CN subs
PQBN |- PQNB or PQBN |- PNQB...BN subs
PNN |- P and P |- PNN...N subs (double negation)

PQARC, RNPA |- QN
1. PQARC
2. RNPA \ show QN
3. RNPQANC.......1 C subs
4. RN..................2 A elim
5. PQAN..............3,4 C elim
6. PNQNO............5 AN subs
7. P.....................2 A elim
8. QN..................6,7 O elim

RPQAC, QN |- RN
1. RPQAC
2. QN \ show RN
3. PQANRNC.......1 C subs
4. PNQNORNC.....3 AN subs
5. PNQNO............2 O int
6. RN..................2,5 O elim

Go on to quantifier rules.

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