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Theorem iftrued 3451
Description: Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
iftrued.1  |-  ( ph  ->  ch )
Assertion
Ref Expression
iftrued  |-  ( ph  ->  if ( ch ,  A ,  B )  =  A )

Proof of Theorem iftrued
StepHypRef Expression
1 iftrued.1 . 2  |-  ( ph  ->  ch )
2 iftrue 3449 . 2  |-  ( ch 
->  if ( ch ,  A ,  B )  =  A )
31, 2syl 14 1  |-  ( ph  ->  if ( ch ,  A ,  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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-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:  eqifdc  3476  mposnif  5833  fimax2gtrilemstep  6762  updjudhcoinlf  6933  omp1eomlem  6947  difinfsnlem  6952  ctssdclemn0  6963  ctssdc  6966  enumctlemm  6967  fodju0  6987  iseqf1olemnab  10216  iseqf1olemab  10217  iseqf1olemqk  10222  iseqf1olemfvp  10225  seq3f1olemqsumkj  10226  seq3f1olemqsum  10228  seq3f1oleml  10231  seq3f1o  10232  fser0const  10244  expnnval  10251  2zsupmax  10952  xrmaxifle  10970  xrmaxiflemab  10971  xrmaxiflemlub  10972  xrmaxiflemcom  10973  summodclem3  11104  summodclem2a  11105  isum  11109  fsum3  11111  isumss  11115  fsumcl2lem  11122  fsumadd  11130  fsummulc2  11172  cvgratz  11256  ef0lem  11280  gcdval  11560  ennnfonelemss  11834  ennnfonelemkh  11836  ennnfonelemhf1o  11837  ressid2  11929  subctctexmid  13092  nninfalllemn  13098  nninfsellemeq  13106  nninfsellemeqinf  13108  nninffeq  13112
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