|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 1692
through ax-7 1535 plus rule
ax-gen 1536). 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 1536. Conditional forms of the converse are given
by ax-12 1633,
ax-15 2102, ax-16 1926, and ax-17 1628.
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 1896.
An interesting alternate axiomatization uses ax467 1752 and ax-5o 1694 in place
of ax-4 1692, ax-5 1533, ax-6 1534,
and ax-7 1535.
This axiom is redundant in the presence of certain other axioms, as shown
by theorem ax4 1691. (We replaced the older ax-5o 1694 and ax-6o 1697 with newer
versions ax-5 1533 and ax-6 1534 in order to prove this redundancy.)
(Contributed by NM, 5-Aug-1993.)