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	$$⊢\neg \forall x\phi \to \forall x\neg \forall x\phi$$

Detailed syntax breakdown

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