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Theorem ax5ALT 34716
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 1991 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 1870, ax-c4 34693, ax-c5 34692, ax-11 2190, ax-c7 34694, ax-7 2093, ax-c9 34699, ax-c10 34695, ax-c11 34696, ax-8 2147, ax-9 2154, ax-c14 34700, ax-c15 34698, and ax-c16 34701: in that system, we can derive any instance of ax-5 1991 not containing wff variables by induction on formula length, using ax5eq 34741 and ax5el 34746 for the basis together with hbn 2311, hbal 2192, and hbim 2291. 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 1991 1 (𝜑 → ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629
This theorem was proved from axioms:  ax-5 1991
This theorem is referenced by: (None)
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