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Axiom ax-6 1875
Description: Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. This axiom tells us is that at least one thing exists. In this form (not requiring that 𝑥 and 𝑦 be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) It is equivalent to axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is established by axc10 2240 and ax6fromc10 33199. A more convenient form of this axiom is ax6e 2238, which has additional remarks.

Raph Levien proved the independence of this axiom from the other logical axioms on 12-Apr-2005. See item 16 at

ax-6 1875 can be proved from the weaker version ax6v 1876 requiring that the variables be distinct; see theorem ax6 2239.

ax-6 1875 can also be proved from the Axiom of Separation (in the form that we use that axiom, where free variables are not universally quantified). See theorem ax6vsep 4713.

Except by ax6v 1876, this axiom should not be referenced directly. Instead, use theorem ax6 2239. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

Ref Expression
ax-6 ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Detailed syntax breakdown of Axiom ax-6
StepHypRef Expression
1 vx . . . . 5 setvar 𝑥
2 vy . . . . 5 setvar 𝑦
31, 2weq 1861 . . . 4 wff 𝑥 = 𝑦
43wn 3 . . 3 wff ¬ 𝑥 = 𝑦
54, 1wal 1473 . 2 wff 𝑥 ¬ 𝑥 = 𝑦
65wn 3 1 wff ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff setvar class
This axiom is referenced by:  ax6v  1876
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