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Axiom ax-i9 1420
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 1397 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.) Another name for this theorem is a9e 1583, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.)
Ref Expression
ax-i9 x x = y

Detailed syntax breakdown of Axiom ax-i9
StepHypRef Expression
1 vx . . 3 setvar x
2 vy . . 3 setvar y
31, 2weq 1389 . 2 wff x = y
43, 1wex 1378 1 wff x x = y
Colors of variables: wff set class
This axiom is referenced by:  ax-9  1421  a9e  1583  a9ev  1584  sbcof2  1688
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