|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 x, it is true for any
(that would typically occur as a free variable in the wff
substituted for φ). (A free variable is one that does
not occur in
the scope of a quantifier: x and y are both free in x = y,
but only x is
free in ∀yx = y.) 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 1335. Conditional forms of the converse are given
by ax-12 1399,
ax-16 1692, and ax-17 1416.
Unlike the more general textbook Axiom of Specialization, we cannot choose
a variable different from x 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 1655.
(Contributed by NM, 5-Aug-1993.)