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Theorem ax5ALT 36161
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 1911 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 1797, ax-c4 36138, ax-c5 36137, ax-11 2161, ax-c7 36139, ax-7 2015, ax-c9 36144, ax-c10 36140, ax-c11 36141, ax-8 2116, ax-9 2124, ax-c14 36145, ax-c15 36143, and ax-c16 36146: in that system, we can derive any instance of ax-5 1911 not containing wff variables by induction on formula length, using ax5eq 36186 and ax5el 36191 for the basis together with hbn 2304, hbal 2174, and hbim 2308. 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 1911 1 (𝜑 → ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536
This theorem was proved from axioms:  ax-5 1911
This theorem is referenced by: (None)
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