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Theorem bdsepnft 13115
Description: Closed form of bdsepnf 13116. Version of ax-bdsep 13112 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 13113 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 13114 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)
3 nfnf1 1523 . . . 4  |-  F/ b F/ b ph
43nfal 1555 . . 3  |-  F/ b A. x F/ b
ph
5 nfa1 1521 . . . 4  |-  F/ x A. x F/ b ph
6 nfvd 1509 . . . . 5  |-  ( A. x F/ b ph  ->  F/ b  x  e.  y )
7 nfv 1508 . . . . . . 7  |-  F/ b  x  e.  a
87a1i 9 . . . . . 6  |-  ( A. x F/ b ph  ->  F/ b  x  e.  a )
9 sp 1488 . . . . . 6  |-  ( A. x F/ b ph  ->  F/ b ph )
108, 9nfand 1547 . . . . 5  |-  ( A. x F/ b ph  ->  F/ b ( x  e.  a  /\  ph )
)
116, 10nfbid 1567 . . . 4  |-  ( A. x F/ b ph  ->  F/ b ( x  e.  y  <->  ( x  e.  a  /\  ph )
) )
125, 11nfald 1733 . . 3  |-  ( A. x F/ b ph  ->  F/ b A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
) )
13 nfv 1508 . . . . . 6  |-  F/ x  y  =  b
145, 13nfan 1544 . . . . 5  |-  F/ x
( A. x F/ b ph  /\  y  =  b )
15 elequ2 1691 . . . . . . 7  |-  ( y  =  b  ->  (
x  e.  y  <->  x  e.  b ) )
1615adantl 275 . . . . . 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 1594 . . . 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 1899 . 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 1329   F/wnf 1436   E.wex 1468  BOUNDED wbd 13040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bdsep 13112
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-cleq 2132  df-clel 2135
This theorem is referenced by:  bdsepnf  13116
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