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Theorem eqifdc 3421
Description: Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.)
Assertion
Ref Expression
eqifdc  |-  (DECID  ph  ->  ( A  =  if (
ph ,  B ,  C )  <->  ( ( ph  /\  A  =  B )  \/  ( -. 
ph  /\  A  =  C ) ) ) )

Proof of Theorem eqifdc
StepHypRef Expression
1 exmiddc 782 . . 3  |-  (DECID  ph  ->  (
ph  \/  -.  ph )
)
2 simpr 108 . . . . . 6  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  ph )  ->  ph )
3 simpl 107 . . . . . . 7  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  ph )  ->  A  =  if ( ph ,  B ,  C ) )
42iftrued 3396 . . . . . . 7  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  ph )  ->  if ( ph ,  B ,  C )  =  B )
53, 4eqtrd 2120 . . . . . 6  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  ph )  ->  A  =  B )
62, 5jca 300 . . . . 5  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  ph )  ->  ( ph  /\  A  =  B )
)
76ex 113 . . . 4  |-  ( A  =  if ( ph ,  B ,  C )  ->  ( ph  ->  (
ph  /\  A  =  B ) ) )
8 simpr 108 . . . . . 6  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  -.  ph )  ->  -.  ph )
9 simpl 107 . . . . . . 7  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  -.  ph )  ->  A  =  if ( ph ,  B ,  C ) )
108iffalsed 3399 . . . . . . 7  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  -.  ph )  ->  if ( ph ,  B ,  C )  =  C )
119, 10eqtrd 2120 . . . . . 6  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  -.  ph )  ->  A  =  C )
128, 11jca 300 . . . . 5  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  -.  ph )  ->  ( -.  ph 
/\  A  =  C ) )
1312ex 113 . . . 4  |-  ( A  =  if ( ph ,  B ,  C )  ->  ( -.  ph  ->  ( -.  ph  /\  A  =  C )
) )
147, 13orim12d 735 . . 3  |-  ( A  =  if ( ph ,  B ,  C )  ->  ( ( ph  \/  -.  ph )  -> 
( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C )
) ) )
151, 14syl5com 29 . 2  |-  (DECID  ph  ->  ( A  =  if (
ph ,  B ,  C )  ->  (
( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C ) ) ) )
16 simpr 108 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  A  =  B )
17 simpl 107 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  ph )
1817iftrued 3396 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  if ( ph ,  B ,  C )  =  B )
1916, 18eqtr4d 2123 . . 3  |-  ( (
ph  /\  A  =  B )  ->  A  =  if ( ph ,  B ,  C )
)
20 simpr 108 . . . 4  |-  ( ( -.  ph  /\  A  =  C )  ->  A  =  C )
21 simpl 107 . . . . 5  |-  ( ( -.  ph  /\  A  =  C )  ->  -.  ph )
2221iffalsed 3399 . . . 4  |-  ( ( -.  ph  /\  A  =  C )  ->  if ( ph ,  B ,  C )  =  C )
2320, 22eqtr4d 2123 . . 3  |-  ( ( -.  ph  /\  A  =  C )  ->  A  =  if ( ph ,  B ,  C )
)
2419, 23jaoi 671 . 2  |-  ( ( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C ) )  ->  A  =  if ( ph ,  B ,  C ) )
2515, 24impbid1 140 1  |-  (DECID  ph  ->  ( A  =  if (
ph ,  B ,  C )  <->  ( ( ph  /\  A  =  B )  \/  ( -. 
ph  /\  A  =  C ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664  DECID wdc 780    = wceq 1289   ifcif 3389
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-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-dc 781  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-if 3390
This theorem is referenced by:  fodjuomnilemm  6780
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