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Theorem bdsepnfALT 13194
Description: Alternate proof of bdsepnf 13193, not using bdsepnft 13192. (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 13191 . 2  |-  E. y A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)
3 nfv 1508 . . . . 5  |-  F/ b  x  e.  y
4 nfv 1508 . . . . . 6  |-  F/ b  x  e.  a
5 bdsepnf.nf . . . . . 6  |-  F/ b
ph
64, 5nfan 1544 . . . . 5  |-  F/ b ( x  e.  a  /\  ph )
73, 6nfbi 1568 . . . 4  |-  F/ b ( x  e.  y  <-> 
( x  e.  a  /\  ph ) )
87nfal 1555 . . 3  |-  F/ b A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)
9 nfv 1508 . . 3  |-  F/ y A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
10 elequ2 1691 . . . . 5  |-  ( y  =  b  ->  (
x  e.  y  <->  x  e.  b ) )
1110bibi1d 232 . . . 4  |-  ( y  =  b  ->  (
( x  e.  y  <-> 
( x  e.  a  /\  ph ) )  <-> 
( x  e.  b  <-> 
( x  e.  a  /\  ph ) ) ) )
1211albidv 1796 . . 3  |-  ( y  =  b  ->  ( A. x ( x  e.  y  <->  ( x  e.  a  /\  ph )
)  <->  A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
) ) )
138, 9, 12cbvex 1729 . 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 144 1  |-  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   A.wal 1329   F/wnf 1436   E.wex 1468  BOUNDED wbd 13117
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 13189
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-cleq 2132  df-clel 2135
This theorem is referenced by: (None)
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