**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
specific (that
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 .) 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 1666. Conditional forms of the converse are given
by ax-13 2054,
ax-c14 32426, ax-c16 32427, and ax-5 1749.
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 2148.
An interesting alternate axiomatization uses axc5c711 32452 and ax-c4 32419 in
place of ax-c5 32418, ax-4 1679, ax-10 1888, and ax-11 1893.
This axiom is obsolete and should no longer be used. It is proved above
as theorem sp 1911. (Contributed by NM, 3-Jan-1993.)
(New usage is discouraged.) |