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Theorem bdsep2 10393
Description: Version of ax-bdsep 10391 with one DV condition removed and without initial universal quantifier. Use bdsep1 10392 when sufficient. (Contributed by BJ, 5-Oct-2019.)
Hypothesis
Ref Expression
bdsep2.1  |- BOUNDED  ph
Assertion
Ref Expression
bdsep2  |-  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
Distinct variable groups:    a, b, x    ph, b
Allowed substitution hints:    ph( x, a)

Proof of Theorem bdsep2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eleq2 2117 . . . . . 6  |-  ( y  =  a  ->  (
x  e.  y  <->  x  e.  a ) )
21anbi1d 446 . . . . 5  |-  ( y  =  a  ->  (
( x  e.  y  /\  ph )  <->  ( x  e.  a  /\  ph )
) )
32bibi2d 225 . . . 4  |-  ( y  =  a  ->  (
( x  e.  b  <-> 
( x  e.  y  /\  ph ) )  <-> 
( x  e.  b  <-> 
( x  e.  a  /\  ph ) ) ) )
43albidv 1721 . . 3  |-  ( y  =  a  ->  ( A. x ( x  e.  b  <->  ( x  e.  y  /\  ph )
)  <->  A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
) ) )
54exbidv 1722 . 2  |-  ( y  =  a  ->  ( E. b A. x ( x  e.  b  <->  ( x  e.  y  /\  ph )
)  <->  E. b A. x
( x  e.  b  <-> 
( x  e.  a  /\  ph ) ) ) )
6 bdsep2.1 . . 3  |- BOUNDED  ph
76bdsep1 10392 . 2  |-  E. b A. x ( x  e.  b  <->  ( x  e.  y  /\  ph )
)
85, 7chvarv 1828 1  |-  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102   A.wal 1257   E.wex 1397  BOUNDED wbd 10319
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038  ax-bdsep 10391
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-cleq 2049  df-clel 2052
This theorem is referenced by:  bdsepnft  10394  bdsepnfALT  10396
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