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Theorem ax5ALT 35923
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 1902 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 1787, ax-c4 35900, ax-c5 35899, ax-11 2151, ax-c7 35901, ax-7 2006, ax-c9 35906, ax-c10 35902, ax-c11 35903, ax-8 2107, ax-9 2115, ax-c14 35907, ax-c15 35905, and ax-c16 35908: in that system, we can derive any instance of ax-5 1902 not containing wff variables by induction on formula length, using ax5eq 35948 and ax5el 35953 for the basis together with hbn 2294, hbal 2164, and hbim 2298. 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.)

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

Proof of Theorem ax5ALT
StepHypRef Expression
1 ax-5 1902 1 (𝜑 → ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526
This theorem was proved from axioms:  ax-5 1902
This theorem is referenced by: (None)
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