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Axiom ax-9 1684
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 1692 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 1814 and ax9from9o 1816. A more convenient form of this axiom is a9e 1817, 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 1684 can be proved from a weaker version requiring that the variables be distinct; see theorem ax9 1683.

ax-9 1684 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 4042. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
ax-9  |-  -.  A. x  -.  x  =  y

Detailed syntax breakdown of Axiom ax-9
StepHypRef Expression
1 vx . . . . 5  set  x
2 vy . . . . 5  set  y
31, 2weq 1620 . . . 4  wff  x  =  y
43wn 5 . . 3  wff  -.  x  =  y
54, 1wal 1532 . 2  wff  A. x  -.  x  =  y
65wn 5 1  wff  -.  A. x  -.  x  =  y
Colors of variables: wff set class
This axiom is referenced by:  ax9v  1685  equidqe  1686  ax4  1691  ax9o  1814  a9e  1817  equid  1818  ax4567to4  26768  ax12-2  27792  ax12-4  27795
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