**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 1606 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 1536, ax-5o 2088,
ax-4 2087, ax-7 1720, ax-6o 2089, ax-8 1661, ax-12o 2094, ax-9o 2090, ax-10o 2091,
ax-13 1698, ax-14 1700, ax-15 2095, ax-11o 2093, and ax-16 2096: in that system,
we can derive any instance of ax-17 1606 not containing wff variables by
induction on formula length, using ax17eq 2135 and ax17el 2141 for the basis
together hbn 1732, hbal 1722, and hbim 1737.
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.) |