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Theorem bj-d0clsepcl 15081
Description: Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-d0clsepcl  |- DECID  ph

Proof of Theorem bj-d0clsepcl
Dummy variables  x  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4145 . . . . . . 7  |-  (/)  e.  _V
21bj-snex 15069 . . . . . 6  |-  { (/) }  e.  _V
32zfauscl 4138 . . . . 5  |-  E. a A. x ( x  e.  a  <->  ( x  e. 
{ (/) }  /\  ph ) )
4 eleq1 2252 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  e.  a  <->  (/)  e.  a ) )
5 eleq1 2252 . . . . . . . 8  |-  ( x  =  (/)  ->  ( x  e.  { (/) }  <->  (/)  e.  { (/)
} ) )
65anbi1d 465 . . . . . . 7  |-  ( x  =  (/)  ->  ( ( x  e.  { (/) }  /\  ph )  <->  ( (/)  e.  { (/)
}  /\  ph ) ) )
74, 6bibi12d 235 . . . . . 6  |-  ( x  =  (/)  ->  ( ( x  e.  a  <->  ( x  e.  { (/) }  /\  ph ) )  <->  ( (/)  e.  a  <-> 
( (/)  e.  { (/) }  /\  ph ) ) ) )
81, 7spcv 2846 . . . . 5  |-  ( A. x ( x  e.  a  <->  ( x  e. 
{ (/) }  /\  ph ) )  ->  ( (/) 
e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) ) )
93, 8eximii 1613 . . . 4  |-  E. a
( (/)  e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) )
101snid 3638 . . . . . . . 8  |-  (/)  e.  { (/)
}
1110biantrur 303 . . . . . . 7  |-  ( ph  <->  (
(/)  e.  { (/) }  /\  ph ) )
1211bicomi 132 . . . . . 6  |-  ( (
(/)  e.  { (/) }  /\  ph )  <->  ph )
1312bibi2i 227 . . . . 5  |-  ( (
(/)  e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) )  <-> 
( (/)  e.  a  <->  ph ) )
1413exbii 1616 . . . 4  |-  ( E. a ( (/)  e.  a  <-> 
( (/)  e.  { (/) }  /\  ph ) )  <->  E. a ( (/)  e.  a  <->  ph ) )
159, 14mpbi 145 . . 3  |-  E. a
( (/)  e.  a  <->  ph )
16 bj-bd0el 15024 . . . . 5  |- BOUNDED  (/)  e.  a
1716ax-bj-d0cl 15080 . . . 4  |- DECID  (/)  e.  a
18 dcbiit 840 . . . 4  |-  ( (
(/)  e.  a  <->  ph )  -> 
(DECID  (/)  e.  a  <-> DECID  ph ) )
1917, 18mpbii 148 . . 3  |-  ( (
(/)  e.  a  <->  ph )  -> DECID  ph )
2015, 19eximii 1613 . 2  |-  E. aDECID  ph
21 bj-ex 14918 . 2  |-  ( E. aDECID 
ph  -> DECID  ph )
2220, 21ax-mp 5 1  |- DECID  ph
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105  DECID wdc 835   A.wal 1362    = wceq 1364   E.wex 1503    e. wcel 2160   (/)c0 3437   {csn 3607
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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pr 4224  ax-bd0 14969  ax-bdim 14970  ax-bdor 14972  ax-bdn 14973  ax-bdal 14974  ax-bdex 14975  ax-bdeq 14976  ax-bdsep 15040  ax-bj-d0cl 15080
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-pr 3614  df-bdc 14997
This theorem is referenced by: (None)
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