Axiom ax-10 2141

Description: Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 2129) 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 2142 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 2141 through ax-13 2380, by invoking ax10w 2129 through ax13w 2136. We encourage proving theorems *without* ax-10 2141 through ax-13 2380 and moving them up to the ax-4 1807 through ax-9 2118 section. (New usage is discouraged.)

Assertion

Ref	Expression
ax-10	⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Detailed syntax breakdown of Axiom ax-10

Step	Hyp	Ref	Expression
1		wph	. . . 4 wff 𝜑
2		vx	. . . 4 setvar 𝑥
3	1, 2	wal 1535	. . 3 wff ∀𝑥𝜑
4	3	wn 3	. 2 wff ¬ ∀𝑥𝜑
5	4, 2	wal 1535	. 2 wff ∀𝑥 ¬ ∀𝑥𝜑
6	4, 5	wi 4	1 wff (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

This axiom is referenced by:  hbn1  2142