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Metamath Proof Explorer

Axiom ax-10

Description: Axiom of Quantified Negation. Axiom C5-2 of Monk2 p. 113. This axiom scheme is logically redundant (see ax10w ) but is used as an auxiliary axiom scheme to achieve scheme completeness. It means that x is not free in -. A. x ph . (Contributed by NM, 21-May-2008) Use its alias hbn1 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 through ax-13 , by invoking ax10w through ax13w . We encourage proving theorems *without* ax-10 through ax-13 and moving them up to the ax-4 through ax-9 section. (New usage is discouraged.)

		Ref	Expression
	Assertion	ax-10	$⊢ \neg \forall x φ \to \forall x \neg \forall x φ$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		vx	$setvar x$
1		wph	$wff φ$
2	1 0	wal	$wff \forall x φ$
3	2	wn	$wff \neg \forall x φ$
4	3 0	wal	$wff \forall x \neg \forall x φ$
5	3 4	wi	$wff (\neg \forall x φ \to \forall x \neg \forall x φ)$