Axiom ax-6 1975

Description: Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. This axiom tells us 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 2385 and ax6fromc10 36555. A more convenient form of this axiom is ax6e 2383, 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 https://us.metamath.org/award2003.html 2383.

ax-6 1975 can be proved from the weaker version ax6v 1976 requiring that the variables be distinct; see Theorem ax6 2384.

ax-6 1975 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 5171.

Except by ax6v 1976, this axiom should not be referenced directly. Instead, use Theorem ax6 2384. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-6	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Detailed syntax breakdown of Axiom ax-6

Step	Hyp	Ref	Expression
1		vx	. . . . 5 setvar 𝑥
2		vy	. . . . 5 setvar 𝑦
3	1, 2	weq 1970	. . . 4 wff 𝑥 = 𝑦
4	3	wn 3	. . 3 wff ¬ 𝑥 = 𝑦
5	4, 1	wal 1540	. 2 wff ∀𝑥 ¬ 𝑥 = 𝑦
6	5	wn 3	1 wff ¬ ∀𝑥 ¬ 𝑥 = 𝑦

This axiom is referenced by:  ax6v  1976