|Description: Axiom of Specialization.
A quantified wff implies the wff without a
quantifier (i.e. an instance, or special case, of the generalized wff).
In other words if something is true for all , it is true for any
would typically occur as a free variable in the wff
substituted for ). (A free variable is one that does not occur in
the scope of a quantifier: and are
both free in ,
but only is free
in .) This is
one of the axioms of
what we call "pure" predicate calculus (ax-4 1271
through ax-7 1217 plus rule
ax-gen 1218). Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint).
Also appears as Axiom B5 of [Tarski] p. 67
(under his system S2, defined
in the last paragraph on p. 77).
Note that the converse of this axiom does not hold in general, but a
weaker inference form of the converse holds and is expressed as rule
ax-gen 1218. Conditional forms of the converse are given
by ax-12 1272,
ax-15 1660, ax-16 1481, and ax-17 1280.
Unlike the more general textbook Axiom of Specialization, we cannot choose
a variable different from for the special case. For use, that
requires the assistance of equality axioms, and we deal with it later
after we introduce the definition of proper substitution - see stdpc4 1451.
The relationship of this axiom to other predicate logic axioms is
different than in the classical case. In particular, the current proof of
ax4 1808 (which derives ax-4 1271
from certain other axioms) relies on ax-3 714
and so is not valid intuitionistically. (Contributed by NM,