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Theorem ifmdc 3477
Description: If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.)
Assertion
Ref Expression
ifmdc  |-  ( A  e.  if ( ph ,  B ,  C )  -> DECID  ph )

Proof of Theorem ifmdc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2178 . . . . 5  |-  ( x  =  A  ->  (
x  e.  if (
ph ,  B ,  C )  <->  A  e.  if ( ph ,  B ,  C ) ) )
21imbi1d 230 . . . 4  |-  ( x  =  A  ->  (
( x  e.  if ( ph ,  B ,  C )  ->  ( ph  \/  -.  ph )
)  <->  ( A  e.  if ( ph ,  B ,  C )  ->  ( ph  \/  -.  ph ) ) ) )
3 df-if 3443 . . . . . 6  |-  if (
ph ,  B ,  C )  =  {
x  |  ( ( x  e.  B  /\  ph )  \/  ( x  e.  C  /\  -.  ph ) ) }
43abeq2i 2226 . . . . 5  |-  ( x  e.  if ( ph ,  B ,  C )  <-> 
( ( x  e.  B  /\  ph )  \/  ( x  e.  C  /\  -.  ph ) ) )
5 simpr 109 . . . . . 6  |-  ( ( x  e.  B  /\  ph )  ->  ph )
6 simpr 109 . . . . . 6  |-  ( ( x  e.  C  /\  -.  ph )  ->  -.  ph )
75, 6orim12i 731 . . . . 5  |-  ( ( ( x  e.  B  /\  ph )  \/  (
x  e.  C  /\  -.  ph ) )  -> 
( ph  \/  -.  ph ) )
84, 7sylbi 120 . . . 4  |-  ( x  e.  if ( ph ,  B ,  C )  ->  ( ph  \/  -.  ph ) )
92, 8vtoclg 2718 . . 3  |-  ( A  e.  if ( ph ,  B ,  C )  ->  ( A  e.  if ( ph ,  B ,  C )  ->  ( ph  \/  -.  ph ) ) )
109pm2.43i 49 . 2  |-  ( A  e.  if ( ph ,  B ,  C )  ->  ( ph  \/  -.  ph ) )
11 df-dc 803 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
1210, 11sylibr 133 1  |-  ( A  e.  if ( ph ,  B ,  C )  -> DECID  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 680  DECID wdc 802    = wceq 1314    e. wcel 1463   ifcif 3442
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-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-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
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-v 2660  df-if 3443
This theorem is referenced by: (None)
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