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Theorem bdsep1 12894
Description: Version of ax-bdsep 12893 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
Hypothesis
Ref Expression
bdsep1.1  |- BOUNDED  ph
Assertion
Ref Expression
bdsep1  |-  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
Distinct variable groups:    a, b, x    ph, a, b
Allowed substitution hint:    ph( x)

Proof of Theorem bdsep1
StepHypRef Expression
1 bdsep1.1 . . 3  |- BOUNDED  ph
21ax-bdsep 12893 . 2  |-  A. a E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
32spi 1499 1  |-  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   A.wal 1312   E.wex 1451  BOUNDED wbd 12821
This theorem was proved from axioms:  ax-mp 5  ax-4 1470  ax-bdsep 12893
This theorem is referenced by:  bdsep2  12895  bdzfauscl  12899  bdbm1.3ii  12900  bj-axemptylem  12901  bj-nalset  12904
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