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 ¬ x φ x ¬ x φ

Detailed syntax breakdown

Step Hyp Ref Expression
0 vx setvar x
1 wph wff φ
2 1 0 wal wff x φ
3 2 wn wff ¬ x φ
4 3 0 wal wff x ¬ x φ
5 3 4 wi wff ¬ x φ x ¬ x φ