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Theorem ifeq1dadc 3472
Description: Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.)
Hypotheses
Ref Expression
ifeq1dadc.1  |-  ( (
ph  /\  ps )  ->  A  =  B )
ifeq1dadc.dc  |-  ( ph  -> DECID  ps )
Assertion
Ref Expression
ifeq1dadc  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)

Proof of Theorem ifeq1dadc
StepHypRef Expression
1 ifeq1dadc.1 . . 3  |-  ( (
ph  /\  ps )  ->  A  =  B )
21ifeq1d 3459 . 2  |-  ( (
ph  /\  ps )  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)
3 iffalse 3452 . . . 4  |-  ( -. 
ps  ->  if ( ps ,  A ,  C
)  =  C )
4 iffalse 3452 . . . 4  |-  ( -. 
ps  ->  if ( ps ,  B ,  C
)  =  C )
53, 4eqtr4d 2153 . . 3  |-  ( -. 
ps  ->  if ( ps ,  A ,  C
)  =  if ( ps ,  B ,  C ) )
65adantl 275 . 2  |-  ( (
ph  /\  -.  ps )  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)
7 ifeq1dadc.dc . . 3  |-  ( ph  -> DECID  ps )
8 exmiddc 806 . . 3  |-  (DECID  ps  ->  ( ps  \/  -.  ps ) )
97, 8syl 14 . 2  |-  ( ph  ->  ( ps  \/  -.  ps ) )
102, 6, 9mpjaodan 772 1  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 682  DECID wdc 804    = wceq 1316   ifcif 3444
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-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-dc 805  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-un 3045  df-if 3445
This theorem is referenced by:  sumeq2  11096  isumss  11128
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