ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-i9 GIF version

Axiom ax-i9 1417
Description: Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. One thing this axiom tells us is that at least one thing exists (although ax-4 1392 and possibly others also tell us that, i.e. they are not valid in the empty domain of a "free logic"). 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 1557 and ax9 1559. A more convenient form of this axiom is a9e 1556, 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 http://us.metamath.org/award2003.html.

ax-9 1418 can be proved from a weaker version requiring that the variables be distinct; see theorem a9wa9 1415.

Assertion
Ref Expression
ax-i9 x x = y

Detailed syntax breakdown of Axiom ax-i9
StepHypRef Expression
1 vx . . 3 set x
2 vy . . 3 set y
31, 2weq 1384 . 2 wff x = y
43, 1wex 1374 1 wff x x = y
Colors of variables: wff set class
This axiom is referenced by:  ax-9  1418  a9e  1556
  Copyright terms: Public domain W3C validator