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Theorem exmidexmid 4088
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 811, peircedc 882, or condc 821.

(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 3150 . . 3  |-  { z  e.  { (/) }  |  ph }  C_  { (/) }
2 df-exmid 4087 . . . 4  |-  (EXMID  <->  A. x
( x  C_  { (/) }  -> DECID  (/) 
e.  x ) )
3 p0ex 4080 . . . . . 6  |-  { (/) }  e.  _V
43rabex 4040 . . . . 5  |-  { z  e.  { (/) }  |  ph }  e.  _V
5 sseq1 3088 . . . . . 6  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( x  C_  {
(/) }  <->  { z  e.  { (/)
}  |  ph }  C_ 
{ (/) } ) )
6 eleq2 2179 . . . . . . 7  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( (/)  e.  x  <->  (/)  e.  { z  e.  { (/)
}  |  ph }
) )
76dcbid 806 . . . . . 6  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  (DECID  (/)  e.  x  <-> DECID  (/)  e.  { z  e.  { (/) }  |  ph } ) )
85, 7imbi12d 233 . . . . 5  |-  ( x  =  { z  e. 
{ (/) }  |  ph }  ->  ( ( x 
C_  { (/) }  -> DECID  (/)  e.  x
)  <->  ( { z  e.  { (/) }  |  ph }  C_  { (/) }  -> DECID  (/)  e.  {
z  e.  { (/) }  |  ph } ) ) )
94, 8spcv 2751 . . . 4  |-  ( A. x ( x  C_  {
(/) }  -> DECID  (/)  e.  x )  ->  ( { z  e.  { (/) }  |  ph }  C_  { (/) }  -> DECID  (/)  e.  {
z  e.  { (/) }  |  ph } ) )
102, 9sylbi 120 . . 3  |-  (EXMID  ->  ( { z  e.  { (/)
}  |  ph }  C_ 
{ (/) }  -> DECID  (/)  e.  { z  e.  { (/) }  |  ph } ) )
111, 10mpi 15 . 2  |-  (EXMID  -> DECID  (/)  e.  { z  e.  { (/) }  |  ph } )
12 0ex 4023 . . . . 5  |-  (/)  e.  _V
1312snid 3524 . . . 4  |-  (/)  e.  { (/)
}
14 biidd 171 . . . . 5  |-  ( z  =  (/)  ->  ( ph  <->  ph ) )
1514elrab 2811 . . . 4  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  (
(/)  e.  { (/) }  /\  ph ) )
1613, 15mpbiran 907 . . 3  |-  ( (/)  e.  { z  e.  { (/)
}  |  ph }  <->  ph )
1716dcbii 808 . 2  |-  (DECID  (/)  e.  {
z  e.  { (/) }  |  ph }  <-> DECID  ph )
1811, 17sylib 121 1  |-  (EXMID  -> DECID  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4  DECID wdc 802   A.wal 1312    = wceq 1314    e. wcel 1463   {crab 2395    C_ wss 3039   (/)c0 3331   {csn 3495  EXMIDwem 4086
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066
This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-exmid 4087
This theorem is referenced by:  exmidn0m  4092  exmid0el  4095  exmidel  4096  exmidundif  4097  exmidundifim  4098  sbthlemi3  6813  sbthlemi5  6815  sbthlemi6  6816  exmidomniim  6979  exmidfodomrlemim  7021
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