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Theorem eqifdc 3569
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 836 . . 3  |-  (DECID  ph  ->  (
ph  \/  -.  ph )
)
2 simpr 110 . . . . . 6  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  ph )  ->  ph )
3 simpl 109 . . . . . . 7  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  ph )  ->  A  =  if ( ph ,  B ,  C ) )
42iftrued 3541 . . . . . . 7  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  ph )  ->  if ( ph ,  B ,  C )  =  B )
53, 4eqtrd 2210 . . . . . 6  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  ph )  ->  A  =  B )
62, 5jca 306 . . . . 5  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  ph )  ->  ( ph  /\  A  =  B )
)
76ex 115 . . . 4  |-  ( A  =  if ( ph ,  B ,  C )  ->  ( ph  ->  (
ph  /\  A  =  B ) ) )
8 simpr 110 . . . . . 6  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  -.  ph )  ->  -.  ph )
9 simpl 109 . . . . . . 7  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  -.  ph )  ->  A  =  if ( ph ,  B ,  C ) )
108iffalsed 3544 . . . . . . 7  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  -.  ph )  ->  if ( ph ,  B ,  C )  =  C )
119, 10eqtrd 2210 . . . . . 6  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  -.  ph )  ->  A  =  C )
128, 11jca 306 . . . . 5  |-  ( ( A  =  if (
ph ,  B ,  C )  /\  -.  ph )  ->  ( -.  ph 
/\  A  =  C ) )
1312ex 115 . . . 4  |-  ( A  =  if ( ph ,  B ,  C )  ->  ( -.  ph  ->  ( -.  ph  /\  A  =  C )
) )
147, 13orim12d 786 . . 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 110 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  A  =  B )
17 simpl 109 . . . . 5  |-  ( (
ph  /\  A  =  B )  ->  ph )
1817iftrued 3541 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  if ( ph ,  B ,  C )  =  B )
1916, 18eqtr4d 2213 . . 3  |-  ( (
ph  /\  A  =  B )  ->  A  =  if ( ph ,  B ,  C )
)
20 simpr 110 . . . 4  |-  ( ( -.  ph  /\  A  =  C )  ->  A  =  C )
21 simpl 109 . . . . 5  |-  ( ( -.  ph  /\  A  =  C )  ->  -.  ph )
2221iffalsed 3544 . . . 4  |-  ( ( -.  ph  /\  A  =  C )  ->  if ( ph ,  B ,  C )  =  C )
2320, 22eqtr4d 2213 . . 3  |-  ( ( -.  ph  /\  A  =  C )  ->  A  =  if ( ph ,  B ,  C )
)
2419, 23jaoi 716 . 2  |-  ( ( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C ) )  ->  A  =  if ( ph ,  B ,  C ) )
2515, 24impbid1 142 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 104    <-> wb 105    \/ wo 708  DECID wdc 834    = wceq 1353   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-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  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-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-if 3535
This theorem is referenced by:  fodjum  7143  nninfwlporlemd  7169  xrmaxiflemcom  11252  subctctexmid  14632
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