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Theorem ax5ALT 34708
 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 1990 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 1869, ax-c4 34685, ax-c5 34684, ax-11 2189, ax-c7 34686, ax-7 2092, ax-c9 34691, ax-c10 34687, ax-c11 34688, ax-8 2146, ax-9 2153, ax-c14 34692, ax-c15 34690, and ax-c16 34693: in that system, we can derive any instance of ax-5 1990 not containing wff variables by induction on formula length, using ax5eq 34733 and ax5el 34738 for the basis together with hbn 2310, hbal 2191, and hbim 2290. 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 1990 1 (𝜑 → ∀𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1628 This theorem was proved from axioms:  ax-5 1990 This theorem is referenced by: (None)
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