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Theorem bj-d0clsepcl 12708
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 3995 . . . . . . 7  |-  (/)  e.  _V
21bj-snex 12692 . . . . . 6  |-  { (/) }  e.  _V
32zfauscl 3988 . . . . 5  |-  E. a A. x ( x  e.  a  <->  ( x  e. 
{ (/) }  /\  ph ) )
4 eleq1 2162 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  e.  a  <->  (/)  e.  a ) )
5 eleq1 2162 . . . . . . . 8  |-  ( x  =  (/)  ->  ( x  e.  { (/) }  <->  (/)  e.  { (/)
} ) )
65anbi1d 456 . . . . . . 7  |-  ( x  =  (/)  ->  ( ( x  e.  { (/) }  /\  ph )  <->  ( (/)  e.  { (/)
}  /\  ph ) ) )
74, 6bibi12d 234 . . . . . 6  |-  ( x  =  (/)  ->  ( ( x  e.  a  <->  ( x  e.  { (/) }  /\  ph ) )  <->  ( (/)  e.  a  <-> 
( (/)  e.  { (/) }  /\  ph ) ) ) )
81, 7spcv 2734 . . . . 5  |-  ( A. x ( x  e.  a  <->  ( x  e. 
{ (/) }  /\  ph ) )  ->  ( (/) 
e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) ) )
93, 8eximii 1549 . . . 4  |-  E. a
( (/)  e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) )
101snid 3503 . . . . . . . 8  |-  (/)  e.  { (/)
}
1110biantrur 299 . . . . . . 7  |-  ( ph  <->  (
(/)  e.  { (/) }  /\  ph ) )
1211bicomi 131 . . . . . 6  |-  ( (
(/)  e.  { (/) }  /\  ph )  <->  ph )
1312bibi2i 226 . . . . 5  |-  ( (
(/)  e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) )  <-> 
( (/)  e.  a  <->  ph ) )
1413exbii 1552 . . . 4  |-  ( E. a ( (/)  e.  a  <-> 
( (/)  e.  { (/) }  /\  ph ) )  <->  E. a ( (/)  e.  a  <->  ph ) )
159, 14mpbi 144 . . 3  |-  E. a
( (/)  e.  a  <->  ph )
16 bj-bd0el 12647 . . . . 5  |- BOUNDED  (/)  e.  a
1716ax-bj-d0cl 12703 . . . 4  |- DECID  (/)  e.  a
18 bj-dcbi 12707 . . . 4  |-  ( (
(/)  e.  a  <->  ph )  -> 
(DECID  (/)  e.  a  <-> DECID  ph ) )
1917, 18mpbii 147 . . 3  |-  ( (
(/)  e.  a  <->  ph )  -> DECID  ph )
2015, 19eximii 1549 . 2  |-  E. aDECID  ph
21 bj-ex 12551 . 2  |-  ( E. aDECID 
ph  -> DECID  ph )
2220, 21ax-mp 7 1  |- DECID  ph
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104  DECID wdc 786   A.wal 1297    = wceq 1299   E.wex 1436    e. wcel 1448   (/)c0 3310   {csn 3474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pr 4069  ax-bd0 12592  ax-bdim 12593  ax-bdor 12595  ax-bdn 12596  ax-bdal 12597  ax-bdex 12598  ax-bdeq 12599  ax-bdsep 12663  ax-bj-d0cl 12703
This theorem depends on definitions:  df-bi 116  df-dc 787  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-sn 3480  df-pr 3481  df-bdc 12620
This theorem is referenced by: (None)
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