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Theorem bdsep1 13629
Description: Version of ax-bdsep 13628 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 13628 . 2  |-  A. a E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
32spi 1523 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 1340   E.wex 1479  BOUNDED wbd 13556
This theorem was proved from axioms:  ax-mp 5  ax-4 1497  ax-bdsep 13628
This theorem is referenced by:  bdsep2  13630  bdzfauscl  13634  bdbm1.3ii  13635  bj-axemptylem  13636  bj-nalset  13639
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