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Theorem iftrue 3484
Description: Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
iftrue  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )

Proof of Theorem iftrue
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-if 3480 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
2 dedlema 954 . . 3  |-  ( ph  ->  ( x  e.  A  <->  ( ( x  e.  A  /\  ph )  \/  (
x  e.  B  /\  -.  ph ) ) ) )
32abbi2dv 2259 . 2  |-  ( ph  ->  A  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) } )
41, 3eqtr4id 2192 1  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 698    = wceq 1332    e. wcel 1481   {cab 2126   ifcif 3479
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 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-if 3480
This theorem is referenced by:  iftruei  3485  iftrued  3486  ifsbdc  3491  ifcldadc  3506  ifbothdadc  3508  ifbothdc  3509  ifiddc  3510  ifcldcd  3512  ifandc  3513  fidifsnen  6772  nnnninf  7031  mkvprop  7040  uzin  9382  fzprval  9893  fztpval  9894  modifeq2int  10190  bcval  10527  bcval2  10528  sumrbdclem  11178  fsum3cvg  11179  summodclem2a  11182  isumss2  11194  fsum3ser  11198  fsumsplit  11208  sumsplitdc  11233  prodrbdclem  11372  fproddccvg  11373  iprodap  11381  iprodap0  11383  flodddiv4  11667  gcd0val  11685  dfgcd2  11738  eucalgf  11772  eucalginv  11773  eucalglt  11774  unct  11991  dvexp2  12884  nnsf  13374  nninfsellemsuc  13383
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