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Theorem bdsepnft 9880
Description: Closed form of bdsepnf 9881. Version of ax-bdsep 9877 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 9878 when sufficient. (Contributed by BJ, 19-Oct-2019.)
Hypothesis
Ref Expression
bdsepnft.1  |- BOUNDED  ph
Assertion
Ref Expression
bdsepnft  |-  ( A. x F/ b ph  ->  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
) )
Distinct variable group:    a, b, x
Allowed substitution hints:    ph( x, a, b)

Proof of Theorem bdsepnft
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 bdsepnft.1 . . 3  |- BOUNDED  ph
21bdsep2 9879 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)
3 nfnf1 1436 . . . 4  |-  F/ b F/ b ph
43nfal 1468 . . 3  |-  F/ b A. x F/ b
ph
5 nfa1 1434 . . . 4  |-  F/ x A. x F/ b ph
6 nfvd 1422 . . . . 5  |-  ( A. x F/ b ph  ->  F/ b  x  e.  y )
7 nfv 1421 . . . . . . 7  |-  F/ b  x  e.  a
87a1i 9 . . . . . 6  |-  ( A. x F/ b ph  ->  F/ b  x  e.  a )
9 sp 1401 . . . . . 6  |-  ( A. x F/ b ph  ->  F/ b ph )
108, 9nfand 1460 . . . . 5  |-  ( A. x F/ b ph  ->  F/ b ( x  e.  a  /\  ph )
)
116, 10nfbid 1480 . . . 4  |-  ( A. x F/ b ph  ->  F/ b ( x  e.  y  <->  ( x  e.  a  /\  ph )
) )
125, 11nfald 1643 . . 3  |-  ( A. x F/ b ph  ->  F/ b A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
) )
13 nfv 1421 . . . . . 6  |-  F/ x  y  =  b
145, 13nfan 1457 . . . . 5  |-  F/ x
( A. x F/ b ph  /\  y  =  b )
15 elequ2 1601 . . . . . . 7  |-  ( y  =  b  ->  (
x  e.  y  <->  x  e.  b ) )
1615adantl 262 . . . . . 6  |-  ( ( A. x F/ b
ph  /\  y  =  b )  ->  (
x  e.  y  <->  x  e.  b ) )
1716bibi1d 222 . . . . 5  |-  ( ( A. x F/ b
ph  /\  y  =  b )  ->  (
( x  e.  y  <-> 
( x  e.  a  /\  ph ) )  <-> 
( x  e.  b  <-> 
( x  e.  a  /\  ph ) ) ) )
1814, 17albid 1506 . . . 4  |-  ( ( A. x F/ b
ph  /\  y  =  b )  ->  ( A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)  <->  A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
) ) )
1918ex 108 . . 3  |-  ( A. x F/ b ph  ->  ( y  =  b  -> 
( A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)  <->  A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
) ) ) )
204, 12, 19cbvexd 1802 . 2  |-  ( A. x F/ b ph  ->  ( E. y A. x
( x  e.  y  <-> 
( x  e.  a  /\  ph ) )  <->  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
) ) )
212, 20mpbii 136 1  |-  ( A. x F/ b ph  ->  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98   A.wal 1241   F/wnf 1349   E.wex 1381  BOUNDED wbd 9805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bdsep 9877
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036
This theorem is referenced by:  bdsepnf  9881
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