**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 1616 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 1546, ax-5o 2136,
ax-4 2135, ax-7 1734, ax-6o 2137, ax-8 1675, ax-12o 2142, ax-9o 2138, ax-10o 2139,
ax-13 1712, ax-14 1714, ax-15 2143, ax-11o 2141, and ax-16 2144: in that system,
we can derive any instance of ax-17 1616 not containing wff variables by
induction on formula length, using ax17eq 2183 and ax17el 2189 for the basis
together hbn 1776, hbal 1736, and hbim 1817.
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 setvar variable not occurring in a wff (as
opposed to just two setvar variables being distinct). (Contributed by
NM, 19-Aug-2017.) (New usage is discouraged.) (Proof modification
discouraged.) |