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Theorem a9eNEW7 29470
Description: At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1566 through ax-14 1729 and ax-17 1626, all axioms other than ax9 1953 are believed to be theorems of free logic, although the system without ax9 1953 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
a9eNEW7  |-  E. x  x  =  y

Proof of Theorem a9eNEW7
StepHypRef Expression
1 ax9NEW7 29468 . 2  |-  -.  A. x  -.  x  =  y
2 df-ex 1551 . 2  |-  ( E. x  x  =  y  <->  -.  A. x  -.  x  =  y )
31, 2mpbir 201 1  |-  E. x  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3   A.wal 1549   E.wex 1550
This theorem is referenced by:  equviniNEW7  29527  equs4NEW7  29533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-12 1950  ax-7v 29442
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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