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Theorem bdsepnfALT 14501
Description: Alternate proof of bdsepnf 14500, not using bdsepnft 14499. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
bdsepnf.nf  |-  F/ b
ph
bdsepnf.1  |- BOUNDED  ph
Assertion
Ref Expression
bdsepnfALT  |-  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 bdsepnfALT
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 bdsepnf.1 . . 3  |- BOUNDED  ph
21bdsep2 14498 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)
3 nfv 1528 . . . . 5  |-  F/ b  x  e.  y
4 nfv 1528 . . . . . 6  |-  F/ b  x  e.  a
5 bdsepnf.nf . . . . . 6  |-  F/ b
ph
64, 5nfan 1565 . . . . 5  |-  F/ b ( x  e.  a  /\  ph )
73, 6nfbi 1589 . . . 4  |-  F/ b ( x  e.  y  <-> 
( x  e.  a  /\  ph ) )
87nfal 1576 . . 3  |-  F/ b A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)
9 nfv 1528 . . 3  |-  F/ y A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
10 elequ2 2153 . . . . 5  |-  ( y  =  b  ->  (
x  e.  y  <->  x  e.  b ) )
1110bibi1d 233 . . . 4  |-  ( y  =  b  ->  (
( x  e.  y  <-> 
( x  e.  a  /\  ph ) )  <-> 
( x  e.  b  <-> 
( x  e.  a  /\  ph ) ) ) )
1211albidv 1824 . . 3  |-  ( y  =  b  ->  ( A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)  <->  A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
) ) )
138, 9, 12cbvex 1756 . 2  |-  ( E. y A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)  <->  E. b A. x
( x  e.  b  <-> 
( x  e.  a  /\  ph ) ) )
142, 13mpbi 145 1  |-  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105   A.wal 1351   F/wnf 1460   E.wex 1492  BOUNDED wbd 14424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-bdsep 14496
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-cleq 2170  df-clel 2173
This theorem is referenced by: (None)
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