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Theorem bj-d0clsepcl 10863
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 3907 . . . . . . 7  |-  (/)  e.  _V
21bj-snex 10847 . . . . . 6  |-  { (/) }  e.  _V
32zfauscl 3900 . . . . 5  |-  E. a A. x ( x  e.  a  <->  ( x  e. 
{ (/) }  /\  ph ) )
4 eleq1 2142 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  e.  a  <->  (/)  e.  a ) )
5 eleq1 2142 . . . . . . . 8  |-  ( x  =  (/)  ->  ( x  e.  { (/) }  <->  (/)  e.  { (/)
} ) )
65anbi1d 453 . . . . . . 7  |-  ( x  =  (/)  ->  ( ( x  e.  { (/) }  /\  ph )  <->  ( (/)  e.  { (/)
}  /\  ph ) ) )
74, 6bibi12d 233 . . . . . 6  |-  ( x  =  (/)  ->  ( ( x  e.  a  <->  ( x  e.  { (/) }  /\  ph ) )  <->  ( (/)  e.  a  <-> 
( (/)  e.  { (/) }  /\  ph ) ) ) )
81, 7spcv 2692 . . . . 5  |-  ( A. x ( x  e.  a  <->  ( x  e. 
{ (/) }  /\  ph ) )  ->  ( (/) 
e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) ) )
93, 8eximii 1534 . . . 4  |-  E. a
( (/)  e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) )
101snid 3427 . . . . . . . 8  |-  (/)  e.  { (/)
}
1110biantrur 297 . . . . . . 7  |-  ( ph  <->  (
(/)  e.  { (/) }  /\  ph ) )
1211bicomi 130 . . . . . 6  |-  ( (
(/)  e.  { (/) }  /\  ph )  <->  ph )
1312bibi2i 225 . . . . 5  |-  ( (
(/)  e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) )  <-> 
( (/)  e.  a  <->  ph ) )
1413exbii 1537 . . . 4  |-  ( E. a ( (/)  e.  a  <-> 
( (/)  e.  { (/) }  /\  ph ) )  <->  E. a ( (/)  e.  a  <->  ph ) )
159, 14mpbi 143 . . 3  |-  E. a
( (/)  e.  a  <->  ph )
16 bj-bd0el 10802 . . . . 5  |- BOUNDED  (/)  e.  a
1716ax-bj-d0cl 10858 . . . 4  |- DECID  (/)  e.  a
18 bj-dcbi 10862 . . . 4  |-  ( (
(/)  e.  a  <->  ph )  -> 
(DECID  (/)  e.  a  <-> DECID  ph ) )
1917, 18mpbii 146 . . 3  |-  ( (
(/)  e.  a  <->  ph )  -> DECID  ph )
2015, 19eximii 1534 . 2  |-  E. aDECID  ph
21 bj-ex 10709 . 2  |-  ( E. aDECID 
ph  -> DECID  ph )
2220, 21ax-mp 7 1  |- DECID  ph
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103  DECID wdc 776   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434   (/)c0 3252   {csn 3400
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-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-nul 3906  ax-pr 3966  ax-bd0 10747  ax-bdim 10748  ax-bdor 10750  ax-bdn 10751  ax-bdal 10752  ax-bdex 10753  ax-bdeq 10754  ax-bdsep 10818  ax-bj-d0cl 10858
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-sn 3406  df-pr 3407  df-bdc 10775
This theorem is referenced by: (None)
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