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Theorem bdsepnfALT 10947
Description: Alternate proof of bdsepnf 10946, not using bdsepnft 10945. (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 10944 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)
3 nfv 1462 . . . . 5  |-  F/ b  x  e.  y
4 nfv 1462 . . . . . 6  |-  F/ b  x  e.  a
5 bdsepnf.nf . . . . . 6  |-  F/ b
ph
64, 5nfan 1498 . . . . 5  |-  F/ b ( x  e.  a  /\  ph )
73, 6nfbi 1522 . . . 4  |-  F/ b ( x  e.  y  <-> 
( x  e.  a  /\  ph ) )
87nfal 1509 . . 3  |-  F/ b A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)
9 nfv 1462 . . 3  |-  F/ y A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
10 elequ2 1643 . . . . 5  |-  ( y  =  b  ->  (
x  e.  y  <->  x  e.  b ) )
1110bibi1d 231 . . . 4  |-  ( y  =  b  ->  (
( x  e.  y  <-> 
( x  e.  a  /\  ph ) )  <-> 
( x  e.  b  <-> 
( x  e.  a  /\  ph ) ) ) )
1211albidv 1747 . . 3  |-  ( y  =  b  ->  ( A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)  <->  A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
) ) )
138, 9, 12cbvex 1681 . 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 143 1  |-  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   A.wal 1283   F/wnf 1390   E.wex 1422  BOUNDED wbd 10870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-bdsep 10942
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-cleq 2076  df-clel 2079
This theorem is referenced by: (None)
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