Axiom ax-6 1964

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 2380 and ax6fromc10 38362. A more convenient form of this axiom is ax6e 2378, 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 2378.

ax-6 1964 can be proved from the weaker version ax6v 1965 requiring that the variables be distinct; see Theorem ax6 2379.

ax-6 1964 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 5297.

Except by ax6v 1965, this axiom should not be referenced directly. Instead, use Theorem ax6 2379. (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 1959	. . . 4 wff 𝑥 = 𝑦
4	3	wn 3	. . 3 wff ¬ 𝑥 = 𝑦
5	4, 1	wal 1532	. 2 wff ∀𝑥 ¬ 𝑥 = 𝑦
6	5	wn 3	1 wff ¬ ∀𝑥 ¬ 𝑥 = 𝑦

This axiom is referenced by:  ax6v  1965