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Theorem bj-d0clsepcl 14948
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 4142 . . . . . . 7  |-  (/)  e.  _V
21bj-snex 14936 . . . . . 6  |-  { (/) }  e.  _V
32zfauscl 4135 . . . . 5  |-  E. a A. x ( x  e.  a  <->  ( x  e. 
{ (/) }  /\  ph ) )
4 eleq1 2250 . . . . . . 7  |-  ( x  =  (/)  ->  ( x  e.  a  <->  (/)  e.  a ) )
5 eleq1 2250 . . . . . . . 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 2843 . . . . 5  |-  ( A. x ( x  e.  a  <->  ( x  e. 
{ (/) }  /\  ph ) )  ->  ( (/) 
e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) ) )
93, 8eximii 1612 . . . 4  |-  E. a
( (/)  e.  a  <->  ( (/)  e.  { (/)
}  /\  ph ) )
101snid 3635 . . . . . . . 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 1615 . . . 4  |-  ( E. a ( (/)  e.  a  <-> 
( (/)  e.  { (/) }  /\  ph ) )  <->  E. a ( (/)  e.  a  <->  ph ) )
159, 14mpbi 145 . . 3  |-  E. a
( (/)  e.  a  <->  ph )
16 bj-bd0el 14891 . . . . 5  |- BOUNDED  (/)  e.  a
1716ax-bj-d0cl 14947 . . . 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 1612 . 2  |-  E. aDECID  ph
21 bj-ex 14785 . 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 1361    = wceq 1363   E.wex 1502    e. wcel 2158   (/)c0 3434   {csn 3604
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pr 4221  ax-bd0 14836  ax-bdim 14837  ax-bdor 14839  ax-bdn 14840  ax-bdal 14841  ax-bdex 14842  ax-bdeq 14843  ax-bdsep 14907  ax-bj-d0cl 14947
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-sn 3610  df-pr 3611  df-bdc 14864
This theorem is referenced by: (None)
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