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Theorem bdsep1 10943
Description: Version of ax-bdsep 10942 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 10942 . 2  |-  A. a E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
32spi 1470 1  |-  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   A.wal 1283   E.wex 1422  BOUNDED wbd 10870
This theorem was proved from axioms:  ax-mp 7  ax-4 1441  ax-bdsep 10942
This theorem is referenced by:  bdsep2  10944  bdzfauscl  10948  bdbm1.3ii  10949  bj-axemptylem  10950  bj-nalset  10953
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