MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-10 Structured version   Visualization version   GIF version

Axiom ax-10 2174
Description: Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 2161) 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 2175 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 2174 through ax-13 2408, by invoking ax10w 2161 through ax13w 2168. We encourage proving theorems *without* ax-10 2174 through ax-13 2408 and moving them up to the ax-4 1885 through ax-9 2154 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 1629 . . 3 wff 𝑥𝜑
43wn 3 . 2 wff ¬ ∀𝑥𝜑
54, 2wal 1629 . 2 wff 𝑥 ¬ ∀𝑥𝜑
64, 5wi 4 1 wff (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Colors of variables: wff setvar class
This axiom is referenced by:  hbn1  2175
  Copyright terms: Public domain W3C validator