**Description: **Axiom to quantify a
variable over a formula in which it does not occur.
Axiom C5 in [Megill] p. 444 (p. 11 of
the preprint). Also appears as
Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of
[Monk2] p. 113.
(This theorem simply repeats ax-17 1626 so that we can include the
following note, which applies only to the obsolete axiomatization.)
This axiom is *logically* redundant in the (logically complete)
predicate calculus axiom system consisting of ax-gen 1555, ax-5o 2213,
ax-4 2212, ax-7 1749, ax-6o 2214, ax-8 1687, ax-12o 2219, ax-9o 2215, ax-10o 2216,
ax-13 1727, ax-14 1729, ax-15 2220, ax-11o 2218, and ax-16 2221: in that system,
we can derive any instance of ax-17 1626 not containing wff variables by
induction on formula length, using ax17eq 2260 and ax17el 2266 for the basis
together hbn 1801, hbal 1751, and hbim 1836.
However, if we omit this axiom,
our development would be quite inconvenient since we could work only
with specific instances of wffs containing no wff variables - this axiom
introduces the concept of a set variable not occurring in a wff (as
opposed to just two set variables being distinct). (Contributed by NM,
19-Aug-2017.) (New usage is discouraged.) (Proof modification
discouraged.) |