Description: Axiom of Quantified
Negation. Axiom C5-2 of [Monk2] p. 113. This
axiom
scheme is logically redundant (see ax10w 2124) 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 2137 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 2136 through ax-13 2383,
by invoking ax10w 2124 through ax13w 2131. We encourage proving theorems
*without* ax-10 2136 through ax-13 2383 and moving them up to the ax-4 1801
through
ax-9 2115 section. (New usage is
discouraged.) |