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Theorem bj-axempty2 13077
Description: Axiom of the empty set from bounded separation, alternate version to bj-axempty 13076. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4049 instead. (New usage is discouraged.)
Assertion
Ref Expression
bj-axempty2  |-  E. x A. y  -.  y  e.  x
Distinct variable group:    x, y

Proof of Theorem bj-axempty2
StepHypRef Expression
1 bj-axemptylem 13075 . 2  |-  E. x A. y ( y  e.  x  -> F.  )
2 dfnot 1349 . . . 4  |-  ( -.  y  e.  x  <->  ( y  e.  x  -> F.  )
)
32albii 1446 . . 3  |-  ( A. y  -.  y  e.  x  <->  A. y ( y  e.  x  -> F.  )
)
43exbii 1584 . 2  |-  ( E. x A. y  -.  y  e.  x  <->  E. x A. y ( y  e.  x  -> F.  )
)
51, 4mpbir 145 1  |-  E. x A. y  -.  y  e.  x
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1329   F. wfal 1336   E.wex 1468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-bd0 12996  ax-bdim 12997  ax-bdn 13000  ax-bdeq 13003  ax-bdsep 13067
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337
This theorem is referenced by: (None)
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