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Theorem bdsepnft 12044
Description: Closed form of bdsepnf 12045. Version of ax-bdsep 12041 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 12042 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 12043 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)
3 nfnf1 1482 . . . 4  |-  F/ b F/ b ph
43nfal 1514 . . 3  |-  F/ b A. x F/ b
ph
5 nfa1 1480 . . . 4  |-  F/ x A. x F/ b ph
6 nfvd 1468 . . . . 5  |-  ( A. x F/ b ph  ->  F/ b  x  e.  y )
7 nfv 1467 . . . . . . 7  |-  F/ b  x  e.  a
87a1i 9 . . . . . 6  |-  ( A. x F/ b ph  ->  F/ b  x  e.  a )
9 sp 1447 . . . . . 6  |-  ( A. x F/ b ph  ->  F/ b ph )
108, 9nfand 1506 . . . . 5  |-  ( A. x F/ b ph  ->  F/ b ( x  e.  a  /\  ph )
)
116, 10nfbid 1526 . . . 4  |-  ( A. x F/ b ph  ->  F/ b ( x  e.  y  <->  ( x  e.  a  /\  ph )
) )
125, 11nfald 1691 . . 3  |-  ( A. x F/ b ph  ->  F/ b A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
) )
13 nfv 1467 . . . . . 6  |-  F/ x  y  =  b
145, 13nfan 1503 . . . . 5  |-  F/ x
( A. x F/ b ph  /\  y  =  b )
15 elequ2 1649 . . . . . . 7  |-  ( y  =  b  ->  (
x  e.  y  <->  x  e.  b ) )
1615adantl 272 . . . . . 6  |-  ( ( A. x F/ b
ph  /\  y  =  b )  ->  (
x  e.  y  <->  x  e.  b ) )
1716bibi1d 232 . . . . 5  |-  ( ( A. x F/ b
ph  /\  y  =  b )  ->  (
( x  e.  y  <-> 
( x  e.  a  /\  ph ) )  <-> 
( x  e.  b  <-> 
( x  e.  a  /\  ph ) ) ) )
1814, 17albid 1552 . . . 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 114 . . 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 1851 . 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 147 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 103    <-> wb 104   A.wal 1288   F/wnf 1395   E.wex 1427  BOUNDED wbd 11969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-bdsep 12041
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-cleq 2082  df-clel 2085
This theorem is referenced by:  bdsepnf  12045
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