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Theorem ifmdc 3574
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 2240 . . . . 5  |-  ( x  =  A  ->  (
x  e.  if (
ph ,  B ,  C )  <->  A  e.  if ( ph ,  B ,  C ) ) )
21imbi1d 231 . . . 4  |-  ( x  =  A  ->  (
( x  e.  if ( ph ,  B ,  C )  ->  ( ph  \/  -.  ph )
)  <->  ( A  e.  if ( ph ,  B ,  C )  ->  ( ph  \/  -.  ph ) ) ) )
3 df-if 3535 . . . . . 6  |-  if (
ph ,  B ,  C )  =  {
x  |  ( ( x  e.  B  /\  ph )  \/  ( x  e.  C  /\  -.  ph ) ) }
43abeq2i 2288 . . . . 5  |-  ( x  e.  if ( ph ,  B ,  C )  <-> 
( ( x  e.  B  /\  ph )  \/  ( x  e.  C  /\  -.  ph ) ) )
5 simpr 110 . . . . . 6  |-  ( ( x  e.  B  /\  ph )  ->  ph )
6 simpr 110 . . . . . 6  |-  ( ( x  e.  C  /\  -.  ph )  ->  -.  ph )
75, 6orim12i 759 . . . . 5  |-  ( ( ( x  e.  B  /\  ph )  \/  (
x  e.  C  /\  -.  ph ) )  -> 
( ph  \/  -.  ph ) )
84, 7sylbi 121 . . . 4  |-  ( x  e.  if ( ph ,  B ,  C )  ->  ( ph  \/  -.  ph ) )
92, 8vtoclg 2797 . . 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 835 . 2  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
1210, 11sylibr 134 1  |-  ( A  e.  if ( ph ,  B ,  C )  -> DECID  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 708  DECID wdc 834    = wceq 1353    e. wcel 2148   ifcif 3534
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-if 3535
This theorem is referenced by: (None)
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