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Theorem bj-d0clsepcl 13292
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 4062 . . . . . . 7  |-  (/)  e.  _V
21bj-snex 13280 . . . . . 6  |-  { (/) }  e.  _V
32zfauscl 4055 . . . . 5  |-  E. a A. x ( x  e.  a  <->  ( x  e. 
{ (/) }  /\  ph ) )
4 eleq1 2203 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  e.  a  <->  (/)  e.  a ) )
5 eleq1 2203 . . . . . . . 8  |-  ( x  =  (/)  ->  ( x  e.  { (/) }  <->  (/)  e.  { (/)
} ) )
65anbi1d 461 . . . . . . 7  |-  ( x  =  (/)  ->  ( ( x  e.  { (/) }  /\  ph )  <->  ( (/)  e.  { (/)
}  /\  ph ) ) )
74, 6bibi12d 234 . . . . . 6  |-  ( x  =  (/)  ->  ( ( x  e.  a  <->  ( x  e.  { (/) }  /\  ph ) )  <->  ( (/)  e.  a  <-> 
( (/)  e.  { (/) }  /\  ph ) ) ) )
81, 7spcv 2782 . . . . 5  |-  ( A. x ( x  e.  a  <->  ( x  e. 
{ (/) }  /\  ph ) )  ->  ( (/) 
e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) ) )
93, 8eximii 1582 . . . 4  |-  E. a
( (/)  e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) )
101snid 3562 . . . . . . . 8  |-  (/)  e.  { (/)
}
1110biantrur 301 . . . . . . 7  |-  ( ph  <->  (
(/)  e.  { (/) }  /\  ph ) )
1211bicomi 131 . . . . . 6  |-  ( (
(/)  e.  { (/) }  /\  ph )  <->  ph )
1312bibi2i 226 . . . . 5  |-  ( (
(/)  e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) )  <-> 
( (/)  e.  a  <->  ph ) )
1413exbii 1585 . . . 4  |-  ( E. a ( (/)  e.  a  <-> 
( (/)  e.  { (/) }  /\  ph ) )  <->  E. a ( (/)  e.  a  <->  ph ) )
159, 14mpbi 144 . . 3  |-  E. a
( (/)  e.  a  <->  ph )
16 bj-bd0el 13235 . . . . 5  |- BOUNDED  (/)  e.  a
1716ax-bj-d0cl 13291 . . . 4  |- DECID  (/)  e.  a
18 dcbiit 825 . . . 4  |-  ( (
(/)  e.  a  <->  ph )  -> 
(DECID  (/)  e.  a  <-> DECID  ph ) )
1917, 18mpbii 147 . . 3  |-  ( (
(/)  e.  a  <->  ph )  -> DECID  ph )
2015, 19eximii 1582 . 2  |-  E. aDECID  ph
21 bj-ex 13138 . 2  |-  ( E. aDECID 
ph  -> DECID  ph )
2220, 21ax-mp 5 1  |- DECID  ph
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104  DECID wdc 820   A.wal 1330    = wceq 1332   E.wex 1469    e. wcel 1481   (/)c0 3367   {csn 3531
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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pr 4138  ax-bd0 13180  ax-bdim 13181  ax-bdor 13183  ax-bdn 13184  ax-bdal 13185  ax-bdex 13186  ax-bdeq 13187  ax-bdsep 13251  ax-bj-d0cl 13291
This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-sn 3537  df-pr 3538  df-bdc 13208
This theorem is referenced by: (None)
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