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Theorem exmidexmid 4022
Description: EXMID implies that an arbitrary proposition is decidable. That is, EXMID captures the usual meaning of excluded middle when stated in terms of propositions.

To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 789, peircedc 858, or condc 787.

(Contributed by Jim Kingdon, 18-Jun-2022.)

Assertion
Ref Expression
exmidexmid  |-  (EXMID  -> DECID  ph )

Proof of Theorem exmidexmid
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3104 . . 3  |-  { z  e.  { (/) }  |  ph }  C_  { (/) }
2 df-exmid 4021 . . . 4  |-  (EXMID  <->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
3 p0ex 4014 . . . . . 6  |-  { (/) }  e.  _V
43rabex 3975 . . . . 5  |-  { z  e.  { (/) }  |  ph }  e.  _V
5 sseq1 3045 . . . . . 6  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( x  C_  {
(/) }  <->  { z  e.  { (/)
}  |  ph }  C_ 
{ (/) } ) )
6 eleq2 2151 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( (/)  e.  x  <->  (/)  e.  { z  e.  { (/)
}  |  ph }
) )
76dcbid 786 . . . . . 6  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  (DECID  (/)  e.  x  <-> DECID  (/)  e.  { z  e.  { (/) }  |  ph } ) )
85, 7imbi12d 232 . . . . 5  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( ( x 
C_  { (/) }  -> DECID  (/)  e.  x
)  <->  ( { z  e.  { (/) }  |  ph }  C_  { (/) }  -> DECID  (/)  e.  {
z  e.  { (/) }  |  ph } ) ) )
94, 8spcv 2712 . . . 4  |-  ( A. x ( x  C_  {
(/) }  -> DECID  (/)  e.  x )  ->  ( { z  e.  { (/) }  |  ph }  C_  { (/) }  -> DECID  (/)  e.  {
z  e.  { (/) }  |  ph } ) )
102, 9sylbi 119 . . 3  |-  (EXMID  ->  ( { z  e.  { (/)
}  |  ph }  C_ 
{ (/) }  -> DECID  (/)  e.  { z  e.  { (/) }  |  ph } ) )
111, 10mpi 15 . 2  |-  (EXMID  -> DECID  (/)  e.  { z  e.  { (/) }  |  ph } )
12 0ex 3958 . . . . 5  |-  (/)  e.  _V
1312snid 3470 . . . 4  |-  (/)  e.  { (/)
}
14 biidd 170 . . . . 5  |-  ( z  =  (/)  ->  ( ph  <->  ph ) )
1514elrab 2769 . . . 4  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  (
(/)  e.  { (/) }  /\  ph ) )
1613, 15mpbiran 886 . . 3  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  ph )
1716dcbii 785 . 2  |-  (DECID  (/)  e.  {
z  e.  { (/) }  |  ph }  <-> DECID  ph )
1811, 17sylib 120 1  |-  (EXMID  -> DECID  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4  DECID wdc 780   A.wal 1287    = wceq 1289    e. wcel 1438   {crab 2363    C_ wss 2997   (/)c0 3284   {csn 3441  EXMIDwem 4020
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-dif 2999  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-exmid 4021
This theorem is referenced by:  exmid0el  4024  exmidel  4025  exmidundif  4026  sbthlemi3  6647  sbthlemi5  6649  sbthlemi6  6650  exmidomniim  6776  exmidfodomrlemim  6806
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