|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
specific x (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.) This is one of the axioms of
what we call "pure" predicate calculus (ax-4 1392
through ax-7 1338 plus rule
ax-gen 1339). 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 1339. Conditional forms of the converse are given
by ax-12 1393,
ax-15 1807, ax-16 1644, and ax-17 1402.
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 1619.
An nice alternate axiomatization uses ax467 1455 and ax-5o 1425 in place of
ax-4 1392, ax-5 1336, ax-6 1337,
and ax-7 1338.
This axiom is redundant in the presence of certain other axioms, as shown
by theorem ax4 1423. (We replaced the older ax-5o 1425 and ax-6o 1428 with newer
versions ax-5 1336 and ax-6 1337 in order to prove this