**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 1793 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 1700, ax-c4 33062,
ax-c5 33061, ax-11 1971, ax-c7 33063, ax-7 1885, ax-c9 33068, ax-c10 33064, ax-c11 33065,
ax-8 1940, ax-9 1947, ax-c14 33069, ax-c15 33067, and ax-c16 33070: in that system,
we can derive any instance of ax-5 1793 not containing wff variables by
induction on formula length, using ax5eq 33110 and ax5el 33115 for the basis
together with hbn 2025, hbal 1973, and hbim 2053.
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.) |