**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-5 1704 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 1618, ax-c4 2215,
ax-c5 2214, ax-11 1842, ax-c7 2216, ax-7 1790, ax-c9 2221, ax-c10 2217, ax-c11 2218,
ax-8 1820, ax-9 1822, ax-c14 2222, ax-c15 2220, and ax-c16 2223: in that system,
we can derive any instance of ax-5 1704 not containing wff variables by
induction on formula length, using ax5eq 2262 and ax5el 2267 for the basis
together hbn 1895, hbal 1844, and hbim 1922.
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 is discouraged.) |