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Theorem iftrue 3449
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 dedlema 938 . . 3  |-  ( ph  ->  ( x  e.  A  <->  ( ( x  e.  A  /\  ph )  \/  (
x  e.  B  /\  -.  ph ) ) ) )
21abbi2dv 2236 . 2  |-  ( ph  ->  A  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) } )
3 df-if 3445 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
42, 3syl6reqr 2169 1  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 682    = wceq 1316    e. wcel 1465   {cab 2103   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-11 1469  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-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-if 3445
This theorem is referenced by:  iftruei  3450  iftrued  3451  ifsbdc  3456  ifcldadc  3471  ifbothdadc  3473  ifbothdc  3474  ifiddc  3475  ifcldcd  3477  ifandc  3478  fidifsnen  6732  nnnninf  6991  mkvprop  7000  uzin  9326  fzprval  9830  fztpval  9831  modifeq2int  10127  bcval  10463  bcval2  10464  sumrbdclem  11113  fsum3cvg  11114  summodclem2a  11118  isumss2  11130  fsum3ser  11134  fsumsplit  11144  sumsplitdc  11169  flodddiv4  11558  gcd0val  11576  dfgcd2  11629  eucalgf  11663  eucalginv  11664  eucalglt  11665  unct  11881  dvexp2  12772  nnsf  13126  nninfsellemsuc  13135
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