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Theorem ifbid 3412
Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
Hypothesis
Ref Expression
ifbid.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
ifbid  |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  A ,  B )
)

Proof of Theorem ifbid
StepHypRef Expression
1 ifbid.1 . 2  |-  ( ph  ->  ( ps  <->  ch )
)
2 ifbi 3411 . 2  |-  ( ( ps  <->  ch )  ->  if ( ps ,  A ,  B )  =  if ( ch ,  A ,  B ) )
31, 2syl 14 1  |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  A ,  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   ifcif 3393
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-in1 579  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-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-if 3394
This theorem is referenced by:  ifbieq1d  3413  ifbieq2d  3415  ifbieq12d  3417  ifandc  3427  fodjuomnilemm  6801  fodjuomnilem0  6802  fodjuomni  6804  nnnninf  6806  0tonninf  9845  1tonninf  9846  sumeq1  10744  isummo  10773  zisum  10774  fisum  10778  isumss  10783  sumsplitdc  10826  nninfalllemn  11898  nninfalllem1  11899  nninfsellemdc  11902  nninfself  11905  nninfsellemeq  11906  nninfsellemqall  11907  nninfsellemeqinf  11908  nninfomni  11911
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