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Axiom ax-10 2016
Description: Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 2003) but is used as an auxiliary axiom scheme to achieve scheme completeness. It means that 𝑥 is not free in ¬ ∀𝑥𝜑. (Contributed by NM, 21-May-2008.) Use its alias hbn1 2017 instead if you must use it. Any theorem in first order logic (FOL) that contains only set variables that are all mutually distinct, and has no wff variables, can be proved *without* using ax-10 2016 through ax-13 2245, by invoking ax10w 2003 through ax13w 2010. We encourage proving theorems *without* ax-10 2016 through ax-13 2245 and moving them up to the ax-4 1734 through ax-9 1996 section. (New usage is discouraged.)
Assertion
Ref Expression
ax-10 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Detailed syntax breakdown of Axiom ax-10
StepHypRef Expression
1 wph . . . 4 wff 𝜑
2 vx . . . 4 setvar 𝑥
31, 2wal 1478 . . 3 wff 𝑥𝜑
43wn 3 . 2 wff ¬ ∀𝑥𝜑
54, 2wal 1478 . 2 wff 𝑥 ¬ ∀𝑥𝜑
64, 5wi 4 1 wff (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Colors of variables: wff setvar class
This axiom is referenced by:  hbn1  2017
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