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Axiom ax-9 1677
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 ax9o 1969 and ax9from9o 2266. A more convenient form of this axiom is a9e 1967, 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-9 1677 can be proved from the weaker version ax9v 1678 requiring that the variables be distinct; see theorem ax9 1968.

ax-9 1677 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 ax9vsep 4418.

Except by ax9v 1678, this axiom should not be referenced directly. Instead, use theorem ax9 1968. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Ref Expression
ax-9 ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Detailed syntax breakdown of Axiom ax-9
StepHypRef Expression
1 vx . . . . 5 set 𝑥
2 vy . . . . 5 set 𝑦
31, 2weq 1663 . . . 4 wff 𝑥 = 𝑦
43wn 3 . . 3 wff ¬ 𝑥 = 𝑦
54, 1wal 1556 . 2 wff 𝑥 ¬ 𝑥 = 𝑦
65wn 3 1 wff ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Colors of variables: wff set class
This axiom is referenced by:  ax9v  1678
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