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Theorem bj-axemptylem 11126
Description: Lemma for bj-axempty 11127 and bj-axempty2 11128. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3930 instead. (New usage is discouraged.)
Assertion
Ref Expression
bj-axemptylem  |-  E. x A. y ( y  e.  x  -> F.  )
Distinct variable group:    x, y

Proof of Theorem bj-axemptylem
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bdfal 11067 . . 3  |- BOUNDED F.
21bdsep1 11119 . 2  |-  E. x A. y ( y  e.  x  <->  ( y  e.  z  /\ F.  )
)
3 bi1 116 . . . 4  |-  ( ( y  e.  x  <->  ( y  e.  z  /\ F.  )
)  ->  ( y  e.  x  ->  ( y  e.  z  /\ F.  ) ) )
4 falimd 1300 . . . 4  |-  ( ( y  e.  z  /\ F.  )  -> F.  )
53, 4syl6 33 . . 3  |-  ( ( y  e.  x  <->  ( y  e.  z  /\ F.  )
)  ->  ( y  e.  x  -> F.  )
)
65alimi 1385 . 2  |-  ( A. y ( y  e.  x  <->  ( y  e.  z  /\ F.  )
)  ->  A. y
( y  e.  x  -> F.  ) )
72, 6eximii 1534 1  |-  E. x A. y ( y  e.  x  -> F.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283   F. wfal 1290   E.wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468  ax-bd0 11047  ax-bdim 11048  ax-bdn 11051  ax-bdeq 11054  ax-bdsep 11118
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291
This theorem is referenced by:  bj-axempty  11127  bj-axempty2  11128
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