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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 (𝜑𝜒)
Assertion
Ref Expression
iftrued (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐴)

Proof of Theorem iftrued
StepHypRef Expression
1 iftrued.1 . 2 (𝜑𝜒)
2 iftrue 3449 . 2 (𝜒 → if(𝜒, 𝐴, 𝐵) = 𝐴)
31, 2syl 14 1 (𝜑 → if(𝜒, 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  10229  iseqf1olemab  10230  iseqf1olemqk  10235  iseqf1olemfvp  10238  seq3f1olemqsumkj  10239  seq3f1olemqsum  10241  seq3f1oleml  10244  seq3f1o  10245  fser0const  10257  expnnval  10264  2zsupmax  10965  xrmaxifle  10983  xrmaxiflemab  10984  xrmaxiflemlub  10985  xrmaxiflemcom  10986  summodclem3  11117  summodclem2a  11118  isum  11122  fsum3  11124  isumss  11128  fsumcl2lem  11135  fsumadd  11143  fsummulc2  11185  cvgratz  11269  ef0lem  11293  gcdval  11575  ennnfonelemss  11850  ennnfonelemkh  11852  ennnfonelemhf1o  11853  ressid2  11945  subctctexmid  13123  nninfalllemn  13129  nninfsellemeq  13137  nninfsellemeqinf  13139  nninffeq  13143
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