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.