Note that under Rules we list the runes
of all the
rules used in the proof. This list says that we used the rewrite rules
CAR-CONS and CDR-CONS, the definitions of the functions NOT,
ENDP and APP, and primitive type reasoning
(which is how we simplified the IF and EQUAL terms).
For what it is worth, IMPLIES and AND are actually
macros
that are expanded into IF expressions before
the proof ever begins. The use of macros is not reported among the rules.