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Theorem ax5ALT 35061
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 1953 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 1839, ax-c4 35038, ax-c5 35037, ax-11 2150, ax-c7 35039, ax-7 2055, ax-c9 35044, ax-c10 35040, ax-c11 35041, ax-8 2109, ax-9 2116, ax-c14 35045, ax-c15 35043, and ax-c16 35046: in that system, we can derive any instance of ax-5 1953 not containing wff variables by induction on formula length, using ax5eq 35086 and ax5el 35091 for the basis together with hbn 2270, hbal 2160, and hbim 2274. 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.)

Ref Expression
ax5ALT (𝜑 → ∀𝑥𝜑)
Distinct variable group:   𝜑,𝑥

Proof of Theorem ax5ALT
StepHypRef Expression
1 ax-5 1953 1 (𝜑 → ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1599
This theorem was proved from axioms:  ax-5 1953
This theorem is referenced by: (None)
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